For spinning class you'll want to buy mountain biking shoes. You should attach cleats for optimal use, but even without they are better than sneakers. Do not buy road shoes because when you finally do decide to attach cleats you wont be able to walk in the gym with them.
I just bought some nike's that I love, but they are narrow. If you have a wide foot I have found shimano (mens) run really wide. They have an introductory level shoe with laces rather than velcro which runs about $54.
Consider cleats, they are great. And spin is a good place to practice with them because you can't fall over!