Pumps - There are pumps for mtbiking, and pumps for road biking. It all has to do with the difference in tyres:
MTB tyres require a greater volume of air, yet run at a lower pressure. Road tyres - however - require substantially higher pressure, and pumps are designed bearing these differences in mind.  MTB pumps typically feature larger-diameter barrels able to compress more air per stroke. Road pumps feature a longer, albeit a more ‘slender’ barrel,  pushing less air - yet at a highr pressure - per stroke.
• Checking the chuck - Choose a pump with a head that quickly converts between Presta (small) and Schrader (large/car) valves.
• Frame pump / Minis - Frame pumps are essential for emergencies on the road and trail, but for everyday use you'll want a floor pump. Mini pumps fit best with a bottle cage bracket.
• Floor pump - Floor pumps are a necessity. They are designed with larger barrels, two-handed handles and a sturdy platform to deliver a large volume of air (under substantial pressure) per stroke. A upper range floorpump would probably see you through your lifetime...
• Suspension pumps - A pump used on certain front suspension forks / rear shocks. With their narrow, yet  long barrels, they can compress up to 300 psi. Quite useful when you need it!

Co2 bombs - Carbon dioxide tyre-inflators inflate a tyre in a flash, but caution is called for as they often cause a blowout. A relatively expensive tool that does add  a bit of weight, but worthwhile to shave 8 minutes of pumping-time when out on the road. The CO2 molecule is smaller than nitrogen (the main component of air) and leaks faster from the tube (which could be of some significance during a long distance endurance event). These ‘bombs’ basically come in 12gram (roadwheels), 16gram (offroad& tandem) and 40gram (3 use) canisters, requiring an additional valve-coupling mechanism. Overall they are well received, but some problems such as the coupling mechanism not puncturing the canister when inserted, and an incompatibility of the CO2 with specific tyres and/or sludge systems, have been reported. It' s good practise to check the bomb’s compatibility with the coupler mechanism prior to your ride, and  -having used a bomb ‘out on the road’ - to check and inflate the tyre with a floor pump when back home. Beware the cold when releasing the bomb…

Presta or Schräder? - Presta valves (skinny) fit all rims, while Schrader valves (fat) only fit Schrader-drilled rims. Schrader valves are supported at most garages’ compressors, and work better with the insertion of a slime system. Schrader valves are more robust, but tend to leak marginally more. Valves should always be kept clean, and Presta (skinny) valves’ bead occasionally tightened!

Correct tyre pressure – As a starting point try the tyre manufacturer's recommended pressure - which you'll find printed on the sidewall - but experiment to find which pressure works and feels best for your individual setup. (The maximum figure stamped on the tyre’s sidewall is simply half the rated pressure the tyre could possibly take, before exploding off the rim.) The tyre’s ‘perfect pressure’ is wholly dependant on the weight it has to carry (that be you!) and cannot really be ‘standardised’. Higher tyre pressures often help reduce the chances of puncture, especially snake-bite, but a ‘softer’ tyre’s potential lower rolling resistance and better traction must be off-set against it. Find your preference. Harder, thicker tyres - particularly in the side-walls - are less prone to punctures, but weigh more. Lighter tyres puncture more. Ultra light tyres & tubbies should only be used on a track, or when you have a team car following with ‘wheels on demand.’

• Road rubber more, knobbies less - The most common mistakes are riding with too little pressure on-road or too much pressure off-road. Road tyres are pumped up to high pressures, but - because they're “skinny” - there's hardly any air inside. Consequently, even if only a little leaks out (most bicycle tubes are made of butyl rubber, which is porous and naturally seeps air), the pressure / volume equations within the tyre are greatly reduced. To prevent this, check tyre pressure before every ride. If you don't, you might be riding on too soft tyres, which could invite trouble.
• Off-road rubber requires lower pressures because they are much wider with considerably more air inside. Unfortunately, the tendency is to over inflate off-road tyres. If you're riding your fat tyres exclusively on pavement and smooth surfaces, inflate them as hard as you like (just don't exceed the manufacturer's maximum recommendation as it is there for a reason.
• Go low - If you're riding off road, use lower pressures in the 35- to 45psi range, (depending on the terrain and your weight). This will increase your control and comfort over trails while improving traction and handling. I f you've been riding off-road on 50 to 60 psi, you'll find a difference in comfort / fatigue.
• Don't go too low, though – I will increase the risk of a puncture (this holds true for road and off-road rubber) as softer tyres pick up more debris which may work into the tyres (piercing the tube), or even pinch the tube (between the rim and an obstacle) cutting it in two places (a  “pinch flat” or “snakebite” puncture). Besides damaging the tube, such impact can bend the rim that is an expensive repair. Under-inflated tyres also lack the sidewall rigidity needed for hard cornering, and wear quicker as a result of increased rolling resistance.
• Road - The roads in the real world aren't that smooth, and the jarring effect of a bumpy ride on over-inflated tyres can affect an energy-sapping ride. Properly inflated tyres will roll over bumpy roads smoother and faster. On ultra-smooth roads, however - when rolling resistance is critical - go as high as 140 psi if your tyres are rated to take it. Stay at the lower end of the pressure zone for comfort on regular and other road surfaces. Rolling resistance also manifests itself in an over inflated tyre having to “lift” the bicycle and rider over every imperfection in the road, whereas a properly inflated ‘pliable’ tyre will smoothen the effect of debris, and increase the speed obtained. Drop tyre pressure a tad in wet conditions, as increasing your patch / footprint can compensate for the resulting loss in grip.
• MTB - Yes, it is possible to ride with knobblys on-road, but it’s not recommended. Rather fit slicks, as the rolling resistance of knobbies over slicks are quite dramatic and will increase with speed. In addition, knobbies have a vibrating effect on-road, which will aid fatigue.
  o Long distance road riding on a mtb definitely requires slicks, but accept that even if the tyre is a bit smoother the bike is still much heavier, and the rider’s positioning less aerodynamic to be really competitive.
  o Mud - as a general principle, a softer, broader tyre will ensure both better grip and greater speed off-road, as the patch / footprint is actually more effective in countering rolling resistance (an over inflated tyre has to “lift” the bicycle and rider over every imperfection in the road, whereas a properly inflated ‘pliable’ tyre will smoothen the effect of debris, and increase the speed obtained). In mud – however - a relatively thinner tyre will slice through the muck and reach the firmer ground more easily, and have the benefit of allowing more clearance between the tyre, frame and shock not clogging up as quick. Go for a 1.95 in the wet, and feel free to use a cooking spray on your tyres prior to racing, as the mud tends to cling less to the tyre.

Road tubes: light v heavy? - High-end tubes can come in at about 50g, a number of midrange tubes range between 75-150g, but various brands weigh in at about 200g per tube. In evaluating the pro’s and cons of the more expensive purchase one will have to accept that once the tyre has been pierced, such intrusion will probably pierce the tube (thick or thin). However, a thinner & softer high-end tyre will increase the occurrence of the dreaded pothole induced ‘snake bite’ puncture, as a result of its relative lack of strength. In addition high-end tubes are constructed from latex, while the heavier and intermediate tubes are made of butyl.  Latex breathes even more than butyl, and high-end tubes will necessitate pumping more often.

Rotational inertia. = Anything that rotates needs to be spun up to speed being accelerated along a straight line, and the lighter the better.  The rotating weight of wheels are very important with bicycles, as the ‘weight’ near the perimeter of a wheel has about twice the stored energy of non-rotating weight when moving. The rotation of cranks, wheel hubs and other parts are of little significance because the radius and speed of rotation are small. Rotating weight resists acceleration (and deceleration), therefore lighter rims, spoke nipples, and tyres allow quicker acceleration. There is no significant difference between rotating and non-rotating weight when at steady speeds or when climbing. You need to go really fast for it to make a difference, but for professional cyclists cycling "on the edge" it constitutes technology they can't do without.

Tyre liners – Tyre liners (strips of plastic that lay between your tube and tyre) help deflect piercing objects and significantly help reduce punctures coming through the tread. They require some patience in set up, refitting and may add to rolling resistance, but give a little bit of tyre bulk to ward off snake-bites.

Sludge  - There are a number of aftermarket tyre sealant applications that work well. They do not always require a tubeless tyre, but compatible rims and some care and patience in fitment is essential. In roadie application it would seem that there could be too much pressure loss before the tyre is effectively sealed (the practical problem is that there is just not a lot of air in a road wheel) but if the puncture can be quickly sealed before pressure has dropped too much, results are good (despite the weight gained).  Top cyclists suggest that training wheels can be sludged and even coupled with a tyre liner, but in serious racing the weight gained reduces speed, and is not advisable. In any event, the sludge does not have an indefinite lifespan, is best replaced every 6 months - throw tube away – and has a very messy effect when punctured with ‘old stuff’, or when the tyre is cut. In a nutshell, sludge limits your ability to swap tyres, but is effective in plugging punctures. See the comments about rotational inertia. (NOTE: Tubes with Presta valves require ‘puncturing’ the tube in order to insert the slime, whereas Shrader valves merely require the valve to be removed.

Tubbys. - These tyres consist out of an outer casing made of a rubber running surface and a material part that is glued onto the tubular compatible rim. The inner inflatable casing is made of thin latex. Some “tubbys” can be inflated to 200PSI and only weigh 135g each. These days’ high-pressure tyres and tubes are widely used on track bikes as tubbied wheels are lighter than clinchers, but whether the weight saving is worth the hassle of gluing is a fair question. Popular opinion loves the weight saving - coupled with a ceramic bearing set it can free up to 6 watts of power! – and the ride quality seems to be marginally better, but all of that that comes at a cost of purchase and the repair difficulty inherent to a glued tyre. Tubbys are perhaps useful for pro riders having the benefit of pre- glued wheels at hand in a support vehicle.

Talcum powder between the rim and the tube can reduce the wear on the tube, and the chances of spoke puncture.

Tubeless tyres (expensive) do not snakebite, as there is no tube to pinch. However, their softer compound mean they are more prone to sidewall cuts, wear and lose pressure faster (up to 10psi/day from 60psi), and they are more difficult to repair when punctured. They do provide better traction as they can be run at a relatively lower pressure. A more expensive option, and punctures are harder to fix out on the road/track/bush.

Punctures

Piercing punctures involve an object (thorns, branches, or glass, for example) penetrating your tyre rupturing the tube. There is typically a single hole on the outer circumference of the tube, usually small and round, but it can be rather large and irregular

Snake-bites occur when the tube is pinched between a rock / pothole / sidewalk on the one side, and your rim on the other.  They are typically two parallel slits along the length of your tube.

Spoke punctures are like piercing punctures, except the offending object is an over-stretched spoke that pokes through the rim tape. The hole is usually a little larger than a piercing puncture, and may be irregular; however, the give-away is that it is on the inner circumference. Replace the offending spoke, or file down the end.

Blow-outs occur when the tyre jumps off the rim and is no longer able to contain the tube. These are instantaneous (although a “bubble” may exist for some time before the tube ruptures) and are easy to identify – your tyre is partially or completely off the rim, and there are tatters of tube hanging out. Blow-outs are often the result of running tyres outside recommended pressures, having worn beads (the bit of wire or Kevlar that holds your tyre onto your rim), a major cut or wear in the side-walls, hitting a stone too hard, or poor technique when replacing a tube.

If you hit stones too hard on a regular basis, try running a slightly higher pressure with wider tyres (tyres up to 2.3” for strong riders). If you generally blow out soon after changing a tyre or tube, take more care to properly seat your tyre onto your rim, as the tube might be trapped between the two.

• When you get a puncture, FIND THE REASON! Run you fingers along the inside of the tyre and pull the tyre liner through your fingers a couple times until you find the thorn or piece of glass. It will save you a puncture-fix just about 10minutes from now…
• If you get a large cut in the side-wall, put a piece of pre-cut tyrewall, a chocolate bar wrapper, a piece of tyre liner, the side of a plastic margarine tub or even heavy cardboard inside the tyre (covering the tyre’s cut), inflate and ride conservatively.
• Patching should be done with great care to ensure the seal. Conventional patching requires up to an hour before the tube can be used. Rather carry a spare tube or two, with a patch kit as additional backup.
• Consider new tubes for your important events, perhaps even a pair of competition only tyres. (This is perhaps more important for road riding - where tyre pressures are much higher, speeds are greater and the consequence of a flat more severe – where a few minutes can be material and a sudden flat at 65km/h just isn’t any fun.

No tube or patchkit?

• Remove the offending tube and stuff your tyre with grass. You will have the bumpiest ride of your life - with awful traction - so take it easy. Fill them as hard as you can and stop every few kilometres to refill as the grass gets milled to dust.
• Cut clean through the tube (at the hole) and knot both ends (twice) of the tube. Put this ‘long’, knotted tube into your tyre, and pad the gap with as much grass as possible. Inflate to a rideable pressure, but please ENSURE THERE WAS ONLY ONE HOLE IN THE TUBE!