Velodrome racers typically leave the brakes off their bikes to make the cycle more lightweight. If your bike has no brakes, don't make the mistake of practicing your technique on a busy city street or a full track. Instead, use a velodrome track when it's empty to help you learn the ropes safely. Steering the bike will be similar to what you're used to, but pedaling and stopping are a little different than a typical freewheel bike.
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Adjust the height of your seat so that at least one foot can touch the ground as you sit on the saddle. You might be able to raise your seat a bit higher later, but for now you want to have access to the ground. If your bike has pedal cages or clip-in pedals, flip over the pedal so that you're using the non-clip side. As you're learning, you don't want your feet stuck in the pedals.
Place the pedals in the position you want them by standing next to the bike, pulling up slightly on the frame to raise the back wheel off the ground, and then pressing the nearest pedal forward to your desired location, using your inside foot. Then step over the bike and place your hands on the handlebars.
Sit on the seat, place your feet on the pedals and begin pedaling slowly. When you want to slow down, pedal slower. Your bike's chain has to be in motion all the time, so you can't simply stop pedaling as you can with a standard "freewheel" bike.
Practice "skipping," a method of stopping by raising your body off the saddle, while at the same time moving the pedals so that they're both horizontal to the ground. As the pedals become horizontal from one another, pull up on the forward pedal and press down on the forward one, called a "kick," which will cause the back wheel to stop moving. After the first skip, allow the pedals to move forward again and back to the horizontal position, so that you can perform a second skip. Depending on your speed, you may have to do this several times to slow down enough to stop. Since a fixed-gear bike's chain has to be in motion all the time, making the chain go backward will cause the wheels to go backward.
Practice "skidding," another method of stopping. When the pedals are horizontal, stand up and jerk the bike to one side, causing the tire to skid and the bike to slow down. Allow the bike to skid as long as is comfortable for you, and then place the lower foot down to stop the bike. This method is harder on the bike and can be painful on your knees, so use it only when you need to stop abruptly.