Definitely stay IN the park for the best overall experience. If you want full hookups Trailer Village RV is the place. The free shuttle stops at the entrance. If you don't need hookups and your RV is small enough, Mather campground is very nice. There's also Desert View at the eastern entrance which is also nice and a quieter area but not far from the main area.
If you are free to go any time of year I'd suggest going in April/May or Sept/Oct. Summer can have hotter days and the family campers will add more population.
Check out the park website for lots of good information on things to do, road information, special alerts and camping information.
We've been to the South Rim many times but now we prefer the laid-back North Rim. For a first-time visitor I'd highly recommend the South Rim.
https://www.nps.gov/grca/index.htm