When you first start spin class, you can get away with regular cross-training shoes or whatever footwear you prefer. Given the fact though that you probably complete around 3,500 to 4,500 pedal revolutions in a typical 45-minute class, you want to maximize your pedal stroke ASAP, if not sooner. You'll need the right kind of clips to match your pedals to allow you to pull on the upward part of your stroke cycle.
You need to learn some basic terminology about clips for spin class -- more properly called clipless pedals and shoes. The pedals on spin bikes, like those on higher-end mountain and road bikes, are actually what are called "clipless" shoe-pedal combinations. "Clipless" refers to the fact that you do not need toe clips, as a cleat in the sole of your cycling shoes snaps directly into a spring-loaded clip on the pedal face.
It's likely, but not 100 percent certain, that your spin class bike pedals will feature Shimano's SPD clipless pedals, which match shoes featuring a small two-hole cleat under the ball of the foot. If you already have mountain bike clipless shoes, you have an automatic match -- these always follow the SPD system, as do certain road bike pedals and most spinning-specific shoes. The other side of the pedal features a platform with toe clips for spinning with regular athletic shoes. The recessed cleat of SPD clipless shoes works well for gyms, as you can walk out of the spin room to various parts of the gym without scouring the club carpet or wood floors.
You may find that your gym uses the three-hole or Look clipless system instead, more commonly found on road bikes. These cleats are much larger and may feature an insert that provides some float, or ability to move your foot slightly from side to side without disengaging.
You need to match your shoe's system to that of the gym where you typically take spin classes. With any luck, you can go early to the class with regular workout shoes and talk to the instructor, taking notes on what kind of clipless shoes to get; don't rely on what the front desk attendants tell you, as they may not always know. Gyms that specialize in catering to all kinds of spinners may have both SPD and Look pedals available, or even allow you to bring your own pedals and swap them out on the spin cycle to match your shoes perfectly.