I lift in bare feet, but I also have a home gym--I'm guessing commercial gyms would frown on this?

I have occasionally lifted in my trainers, which are Asics. Not too bad, but better stability with bare feet. Probably a better choice for lifting would be Chucks (Converse), or something else with a flat sole and maybe high tops for ankle support. If I were in the OP's position, I would wear Chucks to lift, and then switch to Asics (or equivalent) for cardio, just keeping the other pair of shoes handy in a gym bag.