…aka some basic stuff about Pedal + Cleat + Shoe
This is a tricky question because technically, any shoes can work with the Peloton bike as long as they have the proper pedals installed. The pedals on a Peloton, like most indoor bikes can be customized based on your preference. The stock pedals are compatible with only Look Delta cleats. Therefore to use those pedals, your shoes will also need to be compatible with Look Delta cleats. Compatible shoes will need to have three holes on the sole to allow for the cleats to be installed properly.
If your shoes only have two holes they will not work with the stock Peloton pedals. Some shoes allow for either type. Others, like popular brand Tiem Athletic only make shoes that work with SPD. Your option here is to either replace your pedals or select new shoes. Switching out your pedals may sound daunting but is actually quite simple, as shown here by the pelo-infamous John Mills. Schwinn Triple Link Pedals are a popular option because they allow for both types of shoes as well as sneaker cages.
Most gyms and studio bikes typically use SPD compatible pedals. A few (like SoulCycle and Flywheel) have double sided pedals that work with Look Delta and SPD. Ironically, commercial Peloton bikes in hotels typically have pedals with SPD on one side and sneaker cages on the other side, which means your Peloton branded shoes will not work with them.
What is your favorite brand of cycling shoes? Have you tried switching out your pedals?