Tyre problems

Pump

Even if you never have punctures, you will need a pump to top up your tires. Try at home if this works: will the pump head fit the valve or do you need an adapter? And can you inflate the tires hard enough. The maximum operating pressure is usually imprinted upon the casing in bar (atm.) or PSI. (100 PSI equals 7 bars). Pump the tires with a track pump and find out how hard your tires could be. Not enough air in the (rear) increases the rolling resistance, increases wear and makes the occurrence of snakebites much more likely. If you go with a group on tour, you can save some weight and space, as not everybody needs to carry a pump.

Tyre levers

With most combinations of rims and tyres you will need one or two tyre levers. My favorite levers are the yellow plastic ones by Michelin. which not only work okay, but are also easy to find once you've dropped them in the verge. But an inferior steel tyre lever is always better than an inferior plastic one.

Puncture kit

A small piece of emery cloth, a tube of tyre cement (not dried out) and several patches in varying sizes should speak for themselves. However many a cyclist has been able to limp home by using a postage stamp or even a knot in the inner tube (don't forget to bring a spare next time). I try to avoid patching tubes on the road, preferring to carry a few spares and repair the tubes at home or in the tent. But this only works if you know how to remove your wheels.
For the rear wheel (the most difficult one) the procedure is as follows.

shift the chain on the smallest sprocket, i.e. the one the furthest on the right
open the brake: most racing bikes have a button on the brake lever (Campy) or on the brake itself, and with cantilevers and V-brakes you can hang out the brake cable. (if you forget this and force the tyre past the brake pads, make sure you check the alignment before you have sawn your sidewalls in two.
open the wheel skewer by folding the lever open (180 degrees) pull the derailleur rearward and push /slam the wheel down or forward, depending on whether you have vertical or horizontal dropouts.
The wheel will probably be caught by the lower chain run, and can be removed towards the left side after you have lifted the cog out of the chain. With some practice you can do this to a loaded touring bike if you lay the bike on the LH pannier.

If you put the wheel back note that:

you will have to pull the derailleur back to open the chain loop so that you can engage the first cog in the chain
you cannot position the axle in the dropouts if you have accidentally closed the skewer, or if the skewer isn't centered left-right on the axle (that is where those little conical springs are for)
you will never get the axle near the dropouts if the tyre catches in the brake
the correct adjustment for the wheel skewer is finger tight with the lever in line with the axle (adjust with the nut on the opposite end) and closing the skewer a further 90 degrees.
don't forget to close/connect the brake, and check to see whether the chain has dropped from the chain ring.

Inner tube

Always take a couple of spare inner tubes on tour. If the valve gets torn out or the tyre explodes, you can forget about patching. Make sure you take tube that fit. A tube fitted with a Schraeder valve (car valve) is usually to big for a rim drilled for Presta (french) valves. And, on the reverse, a vulcanized Presta valve will not take kindly to being forced through an oversize hole (avoid this by using a thin washer)
And are the valves long enough for the rims used? And don't get fobbed of with racing inner tubes if you run wide tyres. Admittedly, a 21 mm tube fitted in a 35 mm cover will probably not explode on the doorstep of the shop you got it from, but it will fail.

Replacing the inner tube

This seems terribly complicated (for instance when you write down all the steps involved) but a sensible approach helps.

Remove the wheel and deflate the inner tube completely. Push the valve inwards to break it free
The tyre beads are often stuck to the rim: go round and push both beads towards the middle of the rim
Insert two tyre levers 2" left cq right of the valve (Some MTB tyres have so much clearance that bare hands will suffice)
fold the tyre levers over (one at a time), if you need a lot of force, retry after you have pushed the tyre bead back in the rim well
If the tyre bead is still tight, remove one lever (hold on to the other one) and use it to fold back another 1-2"
If you have levered enough of the tyre bead over the rim, the tyre levers will slide out and you can use your fingers to pull rest of the tyre over the rim.
Pull the inner tube out of the casing, push the casing aside and remove the valve from the valve hole
Inflate the inner tube and find the leak. Measure the distance of the leak from the valve hole (two chances) and check the casing for damage or foreign matter. Two narrow slits close apart, indicate a snakebite, often caused by to narrow a tyre for the weight or low inflation.

Fitting

Put some air in the inner tube, just enough for the tube to attain its shape. Push the tyre aside to gain access to the valve hole and insert the valve. Lay the rest of the inner tube in the casing, if the inner tube appears to be to long you put in to much air. Insure that the valve goes through the valve hole straight.
Go round the rim and ease the inner tube over the rim wall in the rim well.
Now fit the tyre bead, starting opposite the valve, and working your way up symmetrically. Use the ball of your thumbs, never a tyre lever! With 8" to go the process becomes very difficult
Go back to where you started, (opposite the valve) and start pushing the tyre bead towards the middle of the rim. Push the valve up, to avoid trapping the tube under the tire bead, let perhaps some more air escape and work the remaining tyre bead over the rim wall.
With extremely tight rim-tyre combinations you might have to repeat the process a few times. Do not use a tyre lever, at the risk of pinching the inner tube.
Push up the valve to ensure the tube isn't trapped by the bead
Put some air in the inner tube if you had to deflate it entirely, pinch the beads together all the way round to check that the inner tube isn't trapped (check both sides) and inflate the tyre. Halfway, check that the tyre is seated properly and doesn't wobble.

Cover

A casing failure is not 100% avoidable if you travel on rough roads. Nor do you need long to split a side wall with a misaligned brake pad. Tears can be temporarily repaired with strong thread and a curved needle, but the tyre will never be as good as new again. (other fixes include wrapping a roll of tape around tyre and inner tube before fitting the assembly back to the rim, or covering holes with a section of another tyre from inside etc.).
On long trips you will have to take a spare tyre. That could be a narrow (folding) emergency tyre, if weight and or space are paramount. Fit this to the front, and move the front tyre to the rear if applicable. Swapping tyres halfway could also help to extend their longevity, but don't do this if the rear tyre has worn very square, or your steering will suffer.
Most modern rims have a retaining groove inside the rim wall and will accept folding tyres with a kevlar bead. These tyres come already boxed or in a plastic bag, ready to throw in the pannier bag. Conventional tyres with a steel bead you will have to fold yourself. Grab the tyre at two opposite sides. Bring your hands together and cross them so that the tyre is divided in three equal circles. Fold the outer circles on top of the middle one and tie the three rings together.

Patching inner tubes

The most durable method is with vulcanizing patches and rubber solution. Glue less patches could be convenient on the trail (if they work at all) but are never as durable as the old fashioned way. Patches are available in many sizes, basically you use the smallest possible. But, because you cannot glue patches over another patch, with several holes close together, you might have to be flexible about this.

Start with finding the hole. Inflate the tube and listen for the hiss, hold the tube close to your lips to feel the rushing air, hold the tube under water or in really stubborn cases use soapy water. Mark the spot with ball-point or crayon, and get out the emery cloth. You will need to remove the outer texture of the tube because it is covered with releasing agent. For the glue to adhere, you'll want a mat finish.
Coat the area with a thin even film of cement
Let it dry thoroughly, a dry film on wet cement is not good enough.
Put some air in the tube, remove the aluminum foil from the patch without touching the sticky surface, and push it firmly on the spot required
Remove the transparent backing only if the patch has cured, if you are in a hurry, install the tube and remove the backing the next time you puncture.