As a RV rookie, stayed in Fishing Bridge first visit. Location is good, but its like being in a can of sardines. If you typically dry camp (no hookups) you are better off staying elsewhere in the park. Though if you need a shower, you can drop by there.
Next two trip started with 3 days at Madison, explored what was around there, and moved out to one of the first come campsites in the park, like Tower/Pebble. Call those wildlife viewing days, because you're up at dawn so you'll see more wildlife on your way to find a new campground
Establish your new site, then you are set for a few more days.
Or you can get a reservation at Canyon if you want to start on the east half of the park, then move west to Norris, a fine first come site on the west side though it is much more of a drive to the thermal areas south beyond Norris. (Thats why I have started at Madison and moved west later)
A side trip I have done a number of times is the drive up to the top of Beartooth (going up from Yellowstone to the summit is much easier than the leg down to Red Lodge on the other side).
Chief Joseph is also quite scenic as well, and can be taken on a day trip to the Buffalo Bill museum in Cody. On that trip, we stayed two nights at a nice NF campsite, Crazy Creek, just south of Chief Joseph, using that as a base for the area. Do take the hike to Crazy Creek, a wonderful cascade across from the campground.
Hikes to recommend.
Fairy falls is nice, and you can take a loop back and get a good look over the basin.
Lone Star hike brings you pretty close to a backcountry geyser.
Have enjoyed every Ranger lead hike that has been 2 or more hours long. Anything less is generally just boardwalk strolls. Check the park newspaper for times and places.
Hidden Falls is a nice hike down in Grand Teton, as would be a float on the snake river through the park (sign up for the first float of the day, more animals)
Since I always come up from the south, I spend two days at Grand Teton first, Gros Ventre almost never fills and is fine. Spend a full day, then take the float first thing in th morning, then head up to Yellowstone, stopping at Lewis Falls and maybe west thumb.