Lots of good advice here! I will re-emphasize that you MUST get a permit and check in w/ rangers if you are hiking in the backcountry (i.e. away from the roads). If you are not easily discouraged, get a copy of Lee Whittlesey's "Death in Yellowstone" - there are a great many instances of tragic encounters people have had in the park with animals, hot springs, etc., and knowledge of these things could help you avoid any such incidents. Bear spray is essential if hiking the backcountry in Yellowstone, and some areas may be closed to hiking due to bear activity; rangers will have that information.
Be aware that the road between Norris and Mammoth is undergoing major reconstruction, which will make walking alongside it extra difficult. For that reason you may want to consider staying in Madison CG rather than Norris (though Norris is just on the edge of the construction zone).
A couple of details. Hitchhiking does occur in the park but is primarily done by concession employees - and folks won't take too kindly to someone pretending to be one. :) Folks do hike along the roadsides, and there are plenty of bikers, but the roads are narrow, often with narrow shoulders, so caution is always advised. Drivers aren't always paying full attention to the road, with all there is to see, and that's especially true when wildlife is spotted.
The CG at Bechler is indeed a NFS CG, located just outside the YNP boundary. It's reached by 18+ miles of gravel road east of Ashton, Idaho, and is not accessible to the rest of Yellowstone except by trails.
Canyon is a good base CG, as is Madison or Norris. Madison is probably the most popular one in the park, though - just FYI. I would agree that Grant is probably not the best choice for what you want to do, especially if you plan to visit the Tetons anyway. Check the park website (www.nps.gov, then follow prompts for Yellwostone) for information under "planning your stay" for CG, services & trail information. There may be some rules about holding a campsite if you are not staying in it overnight (just leaving your rig), and you may want to inquire about that before finalizing plans.