I would definitely wear cycling shorts unless you have the option to bring your own saddle and seatpost (check with the gym, but many spin bikes don't take regular seatposts, and if you do that remember you'll have to adjust your tilt, since the spin bike doesn't necessarily have the same seat angle as your own bike).

The OEM saddles on those things are HORRIBLE (squishy with really wide noses), and the last thing I want to do is be sitting on a seam in regular shorts and making it even worse.

I usually wore a cotton T-shirt when I did trainer classes, though. You're not moving around much, so chafing isn't an issue; you're not going through temperature differentials; and cotton helps reduce the size of the puddle underneath you.