One thing I would definitely advise to do, if you're able, is to walk down the Bright Angel Trail if only for a 1/4 miles or so. You will get an altogether perspective of the canyon.
Also, for walking the fairly flat Rim Trail, you can use the free shuttle and get off and on wherever. It's a convenient way to get around the park.
Drive to the east entrance of the park, too. Lots to see at that area.
For lots of good information poke around on the park's official site. Each national park has a site like this and gives you campground information, road repair info, things to do, etc.
I'd also recommend you stay IN the park for the best experience. Trailer Village is the place if you want hookups. The free shuttle stops right outside the RV park entrance or you can even take a stroll to the Rim for a quiet early morning or evening view.