I wouldn't use paper cups. I've used both paper and styrofoam ones,
but the styrofoam ones tend to tip easily and the paper cups start
looking moldy around the bottom fairly quickly. Plus, the bottom
(where you punch the drain holes) gets damp and weak.

I'd either get the cheap plastic trays & domes from the home
store/Walmart or, if they're not available early enough, use plastic
yogurt containers and a bag (until the plants appear). You can use an
awl to put a small hole in the bottom and then enlarge it with
something like a big nail.

Charlotte