Jackson is my hometown. I was a ranger for three years in Grand Teton and my dad was with the Forest Service. That being said, you can drive TO and THROUGH any park, but the true beauty is off the roads. Yes, it's a slow drive in the summer. The John D. Rockefeller parkway between GTNP and YNP can get bogged down.
Both Flagg Ranch (run by private vendor Grand Teton Lodge Company in the JDR parkway) and Gros Ventre Campground in SE GTNP are good advice. Jackson has a KOA that typically is full of seasonal workers that site hop, but you might get a spot. Colter Bay...meh. Boondocking aka "dispersed camping" is available in NFS lands that surround the park, but not IN the parks. Buffalo Fork west of Moran Junction puts you right into the Bridger-Teton National Forest..
Bridger-Teton National Forest - Camping & Cabinsispersed Camping
Yellowstone lost a lot of trees in the campgrounds during the big fire and so a lot of the forest feel was lost - but the trees are coming back.
Cody (NE Entrance to YNP) has an amazing museum (and a fun nightly rodeo) - truly, truly world class. It's a Whitney/Vanderbilt funded Museum, same family as the NYC museum, and it's truly worth a visit - takes days to really go through it though. Unbelievable art and Indian collections, and the firearm collection.... The Beartooth Highway loop is incredible.. worth doing but leave the trailer in the campground.
West Yellowstone has a Wolf and Grizzly attraction if that's your thing. I wouldn't go that way just for that, but that's a pretty side of the park. Buffalo are on that road frequently. It's true they might dent your 'stream if you pushed through the herd..
They are like cows.
Tour guide here: One of the best ways to see GTNP is on a scenic float trip - 10 miles down the Snake River on a very safe, very comfortable section of river. You are more likely to see wildlife on the river than the roads. Guides are all very educated and you will learn a lot. Whitewater is available south of Jackson, if that's your thing. Can park your rigs at the Craig Thomas Visitor Center at Moose Junction in GTNP and meet your float trip guides there - multiple companies use that meet up. Four hour trip generally speaking. They run throughout the day on the hour.
Jackson also put a lot of money into a bike path from town and up through parts of GTNP so you can now cycle safely on a separate paved path that generally parallels the road. GREAT addition - wish it had been there when I was riding my bike back and forth to work. Bikes are available to rent everywhere, including at Dornan's, next to the Moose Visitor Center - which serves an amazing outdoor breakfast served with a view that's hard to beat. Hard to pull a trailer into Dornan's lot though. Google Dornan's for pics.