The main park roads are all fine for your rig. It won't fit in the parking areas at the main attractions, so go directly to the campground, unhook, and then drive your toad around.
Keep your schedule flexible and allow lots of travel time between points. Bear jams happen all the time, and the traffic is a standstill. It can take hours to drive 20 miles, or breeze right through. Keep your camera handy in the front seat, ready to go. Review your settings to take wildlife shots. You may only have a few seconds to take some good shots before the critter disappears back into the forest. You don't have time to fiddle with the settings.
On the road -- wildlife have the right-of-way. If that herd of bison want to loiter in the middle of the road and not let you by -- you sit. Illegal to honk your horn and try to skirt around them. Rangers can cite for that, and not very smart -- a startled bison that jerks the wrong way can rip the side mirror off your car or major dent your body panel.
Drive slower at night. Animals are dark colored and you cannot see them in the middle of the road until the last second.
There are restaurants, stores, and snack bars in the Park, but you will want to bring a cooler with snacks and drinks. Avoid the tourist prices, the crowds at the established eateries, and you can picnic at any of the designated picnic sites all over the Park. Once we had a herd of elk come by while eating at a picnic table and were all around us.
The only caveat is leaving a cooler in your car if you get out and go hiking. The NPS has done a good job of enforcing the food storage rules and as a result the current population of bears are less attracted to cars as they were in the past. But incidents still happen. A grizzly will not even break sweat ripping your car door open like peeling a banana if he smells food inside. The popular parking areas at the main sites are usually OK. There are enough people coming and going to keep the bears away. If you are the only car at the site, they may get bolder.
One of our favorite drives is the unpaved Blacktail Plateau Road. Not many people drive it. It leads into a bit more primitive and undeveloped section of the Park backcountry. You can bicycle the Park roads, but there is no shoulder along most of them, so you are riding in the traffic lane with diesel exhaust and bus fumes in your face. DW and I are both off-trail hikers and not hesitant at all to wander away from the car into the forest. However, that is not for everyone.
For hiking, there are grizzlies in the Park, and there is a chance of an unpleasant encounter. That chance is very small. The chance of a really cool encounter is greater. If the bear detects you coming while still a distance away, he will either scoot away into the woods or be alert and watch out for you. The bad encounters occur when the bear does not detect you until the last second and you startle him close by. Then it is a coin toss which direction he will run -- at you or away. My couple of close encounters resulted in him fleeing. I would not count on that, though. Bear spray is available in the Park (at tourist prices) or at any any outdoors retailer in the country. You can buy a practice canister of inert material to learn how to use the real stuff before your trip. Mid summer most of the bears head up to the higher elevations in the Park where food is and is cooler.
Best place to see wolves is Lamarr Valley in the NE section of the Park. Dusk and dawn. You may or may not see any.
The Museum complex in Cody is worth a stop when passing through. World class museum. We spent all day. Also the Beartooth Highway is on everyone's list of Most Scenic Drives. That is a bit far for a day trip from Fishing Bridge, but if you can swing it while driving to/from Yellowstone, then it is worth your while.
Cell phone coverage is only at the main sites. So your phone GPS mapping app may or may not work. I use the free Avenza app, and download the free Yellowstone map (its the same map is in the Park brochure). Works with no cell coverage, and the app tracks you as you drive.