Definitely make reservations NOW! You will do a lot of driving anyway, and for a week, I'd recommend staying in no more than 2 CGs, as it takes about 1/2 day to change sites (including driving between the CGs.) As others have pointed out, Fishing Bridge is the only CG inside the park with any hookups at all. If you will need them, you will need to camp there or at a CG outside the park. Fishing Bridge is booked through Xanterra, the lodging concessioner for the park; I believe they also manage reservations for the NPS CGs that take them (Madison, Bridge Bay, Grant Village, Canyon).
For camping outside the park, I'd recommend Grizzly RV in West Yellowstone. It's the closest to the park, very clean & nice, and takes reservations. It's also slightly less expensive than the KOA (less shade, though, I think) and is walkable to town. It's been our preference for several years now. We tried Fishing Bridge last fall and, though the location was unbeatable, the sites are TIGHT. It's really cheek-to-jowl and not all that pleasant, IMO.
If you're spending a week in Yellowstone, why not spend a day or two of that time in the Tetons? They're right next door (literally about 5 miles between them) and the pass for one park is good for both. You may want to consider a couple nights in GTNP as well, and again if you need hookups, Colter Bay RV Park would be your best bet. I believe the NPS CGs for GTNP are not reservable - haven't checked into that, so I may be wrong.