I agree with these two answers. There might be a bit of nuance between the two, though. Normally, I'd say "a new pair of shoes", since the shoes are new to me. However, as I often buy used shoes off of eBay, if I bought a pair of shoes that were completely new, I'd say "a pair of new shoes" to emphasize that the shoes themselves are new, and not just "new to me".

Both phrases are correct. It just depends on whether you want to stress that the PAIR is new or that the SHOES are new. But since shoes usually come in pairs, there is little practical difference. However, as Tom hints, in both cases, "new" can mean "never worn by anybody else" or "new to me." The same goes for a new car. "New" can mean "never used by anyone else" (it is straight from the car factory to the car dealer's lot to you) or "new to me" (it is actually a "used car" but it is new to me. So a "new car" (or a new pair of shoes OR a pair of new shoes) can be both new and used at the same time! How can you specify the diference? In two ways: (a) "brand new" usually means "never used by anyone else" (b) by repeating the adjective "new." For example: A: Hey! I am so excited, I bought a new car! B: New new?! A: (if the car has never been used by anyone else): Yes, new new. Or Yes, brand new, or brand spanking new. or A: (if the car has had a previous owner): Well, it is new to me. -- This use of repeated words can also include adverbs. A: I am drving to the grocery store. B: Now now? A: Yes, right now. (Or: No, later today.) -- Or even nouns: A: We are having fish for dinner. B: Do you mean fish fish or fish? (The person is asking do you mean real fish or something that tastes like fish. Or he/she is asking fresh fish or frozen fish? Actually, any distinction in meaning is possible. But this is getting pretty advanced...)

Generally, people would say "a pair of new shoes" but "a new pair of shoes" is also grammatically correct.

Both mean the same thing, but people usually say (at least where I live) "a new pair of shoes."