We've been there twice both times with large trailers. Our favorite campground is Bridge Bay. Anything on the bottom loop is easy access for there. About 10 minutes north is a huge valley great for wildlife watching.
With the TC, you can stay up in Tower Fall allowing easy drive to anything in the north east corner. Restaurant in Theodore Roosevelt to give the DW a break from cooking.
Check for ages, but believe the boys are old enough to take a horse back ride in the area, if not do a chuck wagon event.
If the temps are decent, check on Firehole Canyon. The water is usually at the right temperature for swimming but depends on water flow if the rangers deem it safe. Not an official swimming area, but last time I was there they did allow it and plenty of parking.
Time wise, if you can go before the weekend prior to July 4. That is the weekend crowds really pick up. Weather is usually pretty nice but you could just as well see a little snow.
Both times we have been there, we've stayed 5 nights in the park and wish we had a few more. Next trip I should have 2 weeks, be in a TC, and can hardly wait.
Last piece of advice, check into the issues with having dogs. They will keep you off a lot of trails and make close up watching of wildlife a bit harder. Just leaving them in the camper may not be a good thing or allowed either.
Good luck and have fun.
2015 Ram 3500 4x4 Crew Cab SRW 6.4 Hemi LB 3.73 (12.4 hand calc avg mpg after 92,000 miles with camper)
2004 Lance 815 (prev: 2004 FW 35'; 1994 TT 30'; Tents)