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A very important, but also the most failure-prone part of a bike. Flat tyre is dangerous - it can knock us down in a turn easily, so we must notice a puncture in time.
How much to inflate your tyres? The pressure must be high enough to prevent direct contact between rim and ground. I'd recommend three bars as a safe minimum, wide balloon tyres (2" or more) can live with less. Upper limit is determined by tyre strength and it's usually marked on it. If you want to ride fast, it's good to stay close to this value. If you don't have a pump with a pressure gauge, forget all this and simply check the pressure by feeling the tyre with your fingers and watching how it squashes under your weight.
Tread choice is simple: if you want to ride fast on dry asphalt, there should be as little of it as possible. Aquaplaning can occur only with much wider tyres and at speeds around 100 km/h or more, so water-draining grooves are not necessary. But if the water gets mixed with mud, it become much more slippery, so I'd recommend at least some basic grooving for everyday cycling. Big knobs are best for snow, mud and other loose surfaces, but they take more effort to spin on paved roads.
Material choice is not that simple because the requirements contradict each other: the softer the rubber, the better the grip, but the shorter the lifetime. Cheap Rubena tyres are on the softer end, they grip very well, but wear down in a year and get punctured by every sharp rock (OK, this may be a little exaggerated). On the other hand, Schwalbe Marathon lasts for ages and its thick lining is hard to pierce, but it slips like hell when wet. One has to find some compromise and adapt the riding style accordingly.
How about the width? The more weight, the rougher terrain, and the better "suspension" we want, the wider tyres are needed. Narrow tyres are light and aerodynamic, but that's of no use if we need to slow down to walking pace on every spot of broken tarmac to prevent pinching inner tubes or knocking the rims. My favourite width is 47 mm (1.75"), it's still quite fast when combined with some smooth road tread and I don't have to fear cobblestones and other treats. Of course, not every fork is wide enough to accept a balloon.
A funny curiosity: one of my tyres once developed five of these cracks on its sidewall:
For unknown reasons, outer layer of cord cracked. If the same happens to you, note that you have last several hundred kilometres before the holes wear through the inner tube.
Here the choice is simple again, inner tubes of different brands don't differ much; just avoid suspiciously cheap ones. The only "prank" a tube ever played on me was one Kenda being ungluable with classic rubber glue, so it went to trash after first puncture. On the other hand, it had lasted for quite long before that.
Inner tube should be of the same size as the outer tyre, or slightly slimmer. Not wider - it wouldn't fit inside. Allowed range of tyre sizes is usually marked on each tube (or at least on the box it comes with).
There are three sorts of valves:
Dunlop (Woods, Velo, DV)
The oldest design. Still in use and retail, but no longer mainstream. Sealing is done by a section of soft rubber hose stuck on a hollow stem of the insert (middle and right on the second photo). The hose tends to deteriorate and crack over time, but on the other hand, it can be easily replaced. More reliable hoseless inserts exist (at the left on the second photo), but they leak a bit, so choose whatever you like. A little disadvantage of a Dunlop is that it doesn't fit through a rim hole without unscrewing the locknut, thus deflating the tube. Another problem is that valves long enough for double-walled rims are very hard to find.
Schraeder (AV, SV)
The most widespread type today, together with Presta. It seals by a tiny rubber plate pressed into its seat via spring. To deflate, depress a pin in the centre or unscrew the whole insert (by the forked end of a metal cap at the top right of the second photo). The pin must be kept depressed during inflation, because a pump can't overcome the spring. The end of the pump must cater for that and it must also contain a one-way valve to prevent air from running back out through the open valve. Some Schraeder valves look like the one pictured, with threaded body and a lockring. Others have rubber-coated body and no locknut, so they tend to hide into the rim when you try to push a pump over them.
The newest bike valve design. It is thinner than the previous two, so that the rim doesn't have to be weakened by a big hole (still compatible with big-holed rims, just a bit wobbly), and is almost always long enough for tall double-walled rims. It has Dunlop thread at the very end, but it needs different pump fitting: deeper, with a seal on the outer edge and a one-way valve as for Schraeder. Presta valve is sealed by something rubbery inside which is held in place by the inner air pressure and a locknut on the outside, which you tighten after the tyre is inflated (and which must be fully loosened during the inflation). The valve can't be disassembled, the outer end of the locking pin is riveted. Be careful during pumping, the pin bends easily.
Rim tape is a thing that goes between the rim and inner tube. It prevents punctures by nipple heads in single-walled rims, or by inner tube buckling into holes of a double-walled one. The first ones don't have to be exceptionally strong, you can easily make them from old inner tubes. Caution: some tapes made of harder plastic may have sharp edges around the valve hole, which can puncture the tube (happened to me twice, then I realized it was not a problem of the tube). A tape for a double-wall rim must be much stronger to withstand the air pressure over the nipple holes; it is made of thick polyurethane or rubber-coated fabric.
First insert one bead of the outer tyre into the rim. Then insert inner tube into it and finally snap the other bead in. The inner tube should be inflated slightly so that it maintains its shape and doesn't crumple. If the tyre bead can't be inserted or removed with a bare hand, we need tyre levers - ideally plastic ones which don't scratch aluminium rims. The rim is deeper in the middle, so pushing one half of a bead to that place helps lifting the other half over the rim edge.
If you are not successful on the first try, don't panic. Fifty punctures later, you'll be able to replace a tyre with your eyes closed :-).