One question that is often asked on forums by those new to fixed is "what chainring and sprocket should I choose?". On low-budget fixie conversions, you'll probably be re-using a chainring (since they're more expensive than sprockets of the same quality) so this simplifies things a little.
A good starting point for road use, in terms of gear inches, is 70". This equates to 48 x 18 (most track chainsets have 48T as default), 42 x 16, 52 x 20 or 53 x 20 on a 700c wheel. These feel subtly different; the bigger chainrings are smoother and less sensitive to your chainring being not-quite-circular, but there is a small weight penalty, mainly from the additional chain links. They all work well enough.
70" may seem quite high if you normally spin derailleur gears at 100rpm or more, and you won't be able to really spin this gear on the flat unless you are very powerful. However, the important thing with a fixie is to choose a gear such that you're not spinning out on the downhills, and let the flats and the climbs look after themselves. Fixies climb well even when they might seem to be hopelessly overgeared.
For velodrome use 48 x 15 is typical for a 250m velodrome or larger. For the very small 1/7 km track at Calshot, 48 x 16 (or equivalent - 48 x 16 happens to be a dreadful choice, as I'll explain below) is recommended.
Some people ride velodrome-sized gears on the road, and it works fine for them. They'll find downhills easier and climbs a bit harder.
For time trials the sky is the limit. Traditionslly, TTs were ridden on a medium gear, which was 72" on a 27" wheel, or the usual 48 x 18. These days, short TTs are more like weightlifting on bikes and gears of up to 125" (56 x 12) are not unheard of. It's about having a gear high enough that you can get power down at all times, even on downhill sections. You don't want your feet to be merely following the pedals in a race..
When choosing chainrings and sprockets, avoid those that are a direct multiple, such as 48 x 16 (48/16 = 3). The power strokes will always be on the same teeth at the poor little sprocket and it will wear unevenly and quickly. Ideally the chainring teeth/sprocket teeth should be an irrational number (hence choosing chainrings with prime-numbered teeth is often recommended). Even on a 48 x 15 setup, a used sprocket will show an uneven wear pattern repeating every third tooth (both 48 and 15 are divisible by 3). None of this happens on bikes with derailleur gears because the chain jumps and slips during gearchanges and the chainring and sprocket are not in lockstep with each other.
If you do skip-stops (not that I know anyone who does), the above is even more important, as it also evens out the wear on the rear tyre.
Try to avoid sprockets smaller than 15T as they are less efficient due to the tight radius of the chain. Sometimes you have no choice, e.g. on an MTB conversion where a chainring larger than 40T would hit the chainstay.
Finally, choose good sprockets as cheaper ones can be noisy, soft or have poor threads which can strip your hub. EAI are probably best, and are quiet out of the box, but Surly and Shimano Dura-Ace are also good. The Shimano ones do need a fair bit of running in until they're quiet.