Short answer: not as many as Air Force.
Long Answer:
I don't know the details on how exactly it works because I haven't done it, but there's a soaring summer training available for mids. Basically, before your 1/C (or sometimes 2/C) year, you take ground school classes for a couple weeks and get some time flying gliders. The consensus I've gotten from people who have done it is that it's a good deal.
Over PROTRAMID (before 2/C year) you WILL fly in at least a T-34 (training turboprop) during Aviation week as well as possibly helicopters during aviation and Marine weeks.
During 2/C year, you can also get PRK or LASIK free-of-charge from the government. They prefer PRK because it doesn't limit your service selection. It's a very simple process: you get an email asking if you're interested, you pop down to Bethesda for a screening, go back for the surgery, and are back in your rack heavily medicated by lunchtime.
Before 1/C year, you can go on an aviation cruise. An aviation cruise is a GOOD DEAL. You are assigned to a squadron with a couple other mids (one of my friends just had one other mid with him at his squadron) and basically hang out and fly with the pilots. My friend was at a VR squadron (transport) and got to fly almost every day. His squadron also set up opportunities for him to go and fly with helicopter, prowler (electronic attack jet), and growler (new electronic attack jet based off of the F/A-18) squadrons.
VT-NA, USNA's aviation club, gets lots of aviation related speakers to come and in the past occasionally took people out to the gliders. I'm not sure if they will still do this in the future ($$).
Naval Aviation is not an extremely hard service selection to get. The biggest hurdle is passing the ASTB, which is pretty easy (seriously: I fell asleep during every testing section and still passed). You can take the ASTB whenever you want, but the catch is you can only take it three times and your most recent score is the one that counts, not your highest.
I'd have to check my numbers, but I think Navy Air (SNA and NFO) was either largest or second largest for the class of 2011 with ~240 or 250 pilot slots and ~70-80 NFO. Generally speaking, if you keep your nose clean (no major honor/conduct offenses) and do okay in school, you can get Navy Air.
If you're interested in Naval Aviation (or military aviation in general) check out Airwarriors.com. It's a site dedicated to Navy/Marine/USCG aviation with a LOT of good info and run by aviators. If you do go over there, take the time to read a lot and search for your questions before you post, though.
