Your search for spin shoes will be a 2-part process. First, since you do know that the spin bikes at your gym use SPD pedals, you can go and buy the SPD cleats now. Any reputable bike shop should have them, or you'll be able to find them in hundreds of places online. They run about $20 a pair...and you'll be looking for Shimano SH-51 (single release) SPD cleats.
Next, you'll need to find compatible shoes. Most likely these will be MTB shoes, since MTB shoes use the 2-bolt SPD pattern exclusively. You will find "indoor" spin shoes that look almost like regular sneakers, but I really recommend going with a more rigid shoe. A more flexible shoe may end up causing "hot spots" on your feet. Your best bet would be to try a bunch of different shoes on because sizing can be all over the place with cycling shoes. Once you find a pair that you like, you just install the cleats on them (really simple to do, will take like 3 minutes). I think you'll find that wearing spin shoes in a class will really increase the power to the pedals. I hated those silly "cages" on the spin bikes before I started using cycling shoes. My feet would burn and get hot spots and I was always afraid that the straps would come loose on me. With cycling shoes, you don't have to worry about that and can focus more on the pedal stroke.
2012 Seven Axiom SL - Specialized Ruby SL 155