All spin bikes use SPD compatible cleats.
I prefer MTB or Touring shoes so you can easily walk around the gym.
Towel to drape over handle bars and wipe sweat.
Water bottle. I plan on one bottle per class or hour of work.
Something to clean the bike. I use the disposable disinfecting towels. Most gyms will have paper towels and a cleaner dispenser of some kind. Whatever you are comfortable with, but expect that the person who previously used the bike may not have cleaned it well.
Bike Gloves are a good idea too, some bars are hard metal, also gloves also help to control sweat.
Head sweat band, skull cap or hat.
Plastic or grocery bag to transport wet clothes home. I hang my stuff to dry at home.
I like a long sleeve shirt to help control sweat and not leave a puddle under the bike. You can also lay some paper towels on the floor. Some guys like to have a big puddle under the bike, I guess to gauge performance?
Also, don’t forget, ask at the front desk, if there is a sign-up sheet for the spin classes. Note: some clubs have an additional fee for spin classes.