It kind of depends on what the shoes are made of, (i.e. cloth/leather/synthetic), and what kind of design/ effect you want to get, but pretty much any kind of paint would work. I definitely wouldn't suggest oil because it is expensive and pretty toxic. Acrylic or spraypaint would work, or even gouache. If you have canvas shoes, fabric ink would probably work the best.
when i painted my shoes i used little mettallic paint pens, it worked really well because you can use way more detail and then afterwords i sprayed it with leather shoe protectent so it wouldnt scuff off.