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A4 being a paper size, not a kind of paper. Just an FYI for anyone that doesn't know.

Thus "regular" and "not glossy" (i.e., office paper, 75g).

I use the glossy stuff. Looks fan-freakin'-tastic. Better than retail, honestly.

I print lots of DVD covers (same thing, basically) on regular old printer paper. They don't look half bad, and can look really good if you use a high end printer. My brother printed a few DVD labels out on a really cheap photo paper that wasn't too glossy, and they looked even better.

If you have, or have access to, a decent printer I'd say try regular paper first. If it doesn't look good enough for you, try a photo paper. The only thing is that photo paper can get expensive.

I use 80 lb legal matte paper for all my covers, and they turn out very nice.

So, I am going to have my dad print a cover on normal paper, so that it looks nice until I get a better one, then I will take it to the print shop to do it for me. So this is what the paper should be, correct?
1.Glossy
2.Thin

Generally, yes. Perhaps keep the torn cover behind the replacement for now and take it to the print place when you go. This way they can pick a paper that's similiar and they're less likely to give you a funny look if they're finicky on copyright if they see you really did tear it.