You pick up the IllumiNations cruise at the Yacht Club marina. Officially, you will set sail 30 minutes before the fireworks, but in my experience (we've done it 3 or 4 times), your Captain will usually put in an appearance early, so I'd suggest being there 45-60 minutes ahead of time.
You will then be taken for a sail around Crescent Lake, with your Captain providing information and trivia, before mooring up under the International Gateway bridge (between the UK and France) with the other 5 boats. To be honest, there are better places from which to watch IllumiNations. This location is a bit too far away to get the full impact of the globe, but it is
without question a very special experience. You really do feel like VIPs, especially as most of the people staring at you from above will have no idea of the existence of the cruises

.
I haven't done a Wishes cruise (note to self: why is that?

), but my understanding is that they operate from all the MK resorts (Grand Floridian, Contemporary, Polynesian and Wilderness Lodge). You'll be taken for a cruise around Bay Lake and will get to see the Electrical Water Pageant before moving on to the Seven Seas Lagoon to take up your position in front of the park. You have to pay extra to have the music piped in, but I would have thought that would be a no-brainer.
You pay for the boat, not per person, so you can certainly have it to yourselves as long as you can secure a booking. I've never been successful in doing so, so we've always had to share with other DISers who've been kind enough to offer spaces on their cruises. There is a sticky thread on the Theme Parks forum which gives tips on how to maximise your chances of getting an IllumiNations cruise. I've a feeling the Wishes cruises might be a bit easier to get.