Basic track riding is easy, and it'll be explained as they plonk you on their bike and aim you out onto the track.
Fixed gear - while the wheels are rotating, so are the pedals. They'll strap your feet onto the pedals (or clip them) with the bike stationary and you holding onto the fence or something solid (like a nervous husband), so this isn't scary. A push off and you're riding.
Once you've got some speed up, it's easy to keep rolling. Thing is though, you're used to a freewheel and as soon as you try to coast, the pedal moving up will give that leg a shove - feels like you've hit an ejector seat button but it isn't scary or dangerous, it'll just give you a fright and, if you're normal, this'll happen a couple of times on your first ride, especially as you start to get tired. All part of the fun.
Stopping. Just take the power out of your pedalling and let the bike slow down. As a pedal goes over the top of its stroke, you can use your thighs to put a little bit of back pressure on the pedals to slow you down even more. Don't forget, a velodrome is also banked so as you approach your stopping point, aim up the slope to the fence and gravity will help again. As the bike is about to stop, grab the fence (not too early or the fence will try to disappear behind you).
The thing about the track is that everyone is going the same way and no-one has brakes - they are forbidden. This means that you will never have to do an emergency stop - if someone falls, you just ride around them and no-one can do a sudden stop or suddenly slow down in front of you.
It's all a lot easier than it sounds, and its fun.
Riding a fixie on the track is sheer smoothness. The legs rotate. The bike moves. Everyone else blasts past you like you're standing still (or in my case, the ratbag son tucks onto your rear wheel and then wonders why you're puffing and he isn't). It's fun.
So, when are you going to buy an old road bike and convert it to a fixie for the road?
Richard