OK, so based on traveling Jackson Hole to West Yellowstone, 5 nights there and then back to JH, here is a starting point itinerary for you.
Day 1: Leave Jackson area and enter South gate stopping at waterfalls and overlooks along the way. Stop at West Thumb, Keppler Cascades and then Old Faithful area. I would go to the Visitor Center and kind of get oriented to the area, but I would leave big exploration for another day unless one of the awesome predictable geysers is going to go in the next few hours, then I would try to see it. If you are up for a short hike, I would go up to Observation Point. (Go to the right from Old Faithful, cross bridge, then you will see the sign and trail. It is short but it is a climb!) As you head to West Yellowstone, be sure to take the Firehole Lake Rd on your right which is just north of the Midway Geyser Basin. Overnight in West Yellowstone. If you get to Old Faithful for lunch, I would follow the first signs you see for *gas* and head for the Lower General Store. You will park on the front side (geyser basin side) of the Old Faithful Inn and the Lower Gen. store is right there. You avoid a lot of the traffic that way. You can have lunch, then walk over to the Visitor Center.
Day 2: Make this your geyser day! Head in to the park, turning right at Madison Junction. After you cross the Gibbon River Bridge, turn right into Firehole Canyon Dr. Rejoin the Loop Road. Head to Old Faithful. I suggest you go directly to Old Faithful and skip the geyser basins you will pass…you can pick them up on your way back out to West Yellowstone. Check in at the Visitor Center and write down the eruption times for the predictable geysers on the brochure available at the desk. (50 cents) Try to work your walk thru the basin so you can see some of these geysers. I usually start by going behind Old Faithful on the path to the left side of this geyser, crossing the bridge behind Old Faithful (you can’t see it from the front of the geyser) and up onto Geyser Hill. From there I head down the geyser basin to Morning glory pool (pit toilet here!) and go back toward Old Faithful on the paved path. I like to eat lunch at the Lower General Store. It will be the first store you come to on the right (well, the geysers are on the left, lol)before the Old Faithful Inn as you head back to the Visitor Center. As you leave the Old Faithful area, you can stop at Black Sand Basin, Biscuit Basin and Midway Geyser Basin as your time and energy allow. Take the right turn past Midway into Firehole Lake Dr. (This is where Great Fountain Geyser is located. If there are people around, stop! It is just awesome to watch.) Rejoin the Loop road. You can’t actually go straight across to Fountain Paint Pots, so you can turn left or right and go to the next pull out and turn around. Do Fountain Paint Pots and then head toward Madison and West Yellowstone. ( I usually make a pit stop at Madison just because one time, many years ago, I got caught in a huge bison jam for maybe 45 minutes and did NOT think I was going to make it to West Yellowstone!)Overnight in West Yellowstone
Day 3: Make your way to Canyon. Stop at the Visitor Center. (Nice flush toilets, lol) Head south on the Loop Road, doing the North rim viewpoints (if only one, I like Brink of the Upper Falls) and the South rim. Return to the Loop Road. Go South through Hayden Valley stopping if you see people out with cameras with long lenses or spotting scopes. Continue South picking up Mud Volcano, then go to Fishing Bridge. Park and walk across Fishing Bridge as this area is rich in wildlife. I’ve been able to spot otters, golden eagle, muskrat, white pelican from here. Head back north and do LeHardy Rapids, then north again thru Hayden and back to Canyon. Depending on how your day is going, you might have time to do Norris and Artist Paint Pots. Overnight in West Yellowstone
Day 4: Today will be a fairly long driving day. Go to Canyon. (Pit stop at the VC if needed…flush toilets, lol) Head North over Dunraven stopping at Tower Fall and some of the overlooks north of there. At Tower Roosevelt Jct, turn right in to Lamar Valley. Again look for wildlife or long camera lenses. Turn around and head back to Tower Roosevelt Jct. At Roosevelt, head toward Mammoth. Do the drive back to Petrified Tree, then do Blacktail Plateau (this is dirt/gravel so your rental car company may or may not allow it. It usually is fine in a regular car, no 4WD needed). Many see wildlife back in here, alas I never do! When you get to Mammoth walk the lower terraces. The good bathrooms here are the new ones by the bus parking lot and the gas station. Drive the upper terraces. If you want you can pick up a self guiding tour at the VC and look at the Ft Yellowstone buildings. From Mammoth head back south to Norris. Stop here if you have the time. Overnight in West Yellowstone
Day 5: This is your day to return to anything that needs a second look or to see something you missed on your other days. Return to Jackson for overnight.
There are many hikes and ranger-led programs that you can work in to your days. I encourage you to look at the ranger-led programs online, then see which ones you might want to do. I always do the ranger program at Old Faithful (usually starts at 830 so you have to get away from West Yellowstone by 7 to park, pee and position! lol). The programs are ~usually~ the same year to year. They will be listed in your park newspaper you get when you enter the gate.
There are awesome hikes everywhere in the park. You could do Lone Star trail on your Day 1. The trailhead is located near Keppler Cascades between West Thumb and Old Faithful which you will pass on your way in. You will walk plenty on your geyser day, someone posted on a trip report earlier this summer that they had managed to hike 10 miles on the boardwalks in the geyser basins! You can hike Mt Washburn on your Canyon day, or head down to Storm Point near Fishing Bridge (there is a ranger led hike here this year) or to Natural Bridge just past Bridge Bay marina. There is a hike I want to do on the West Entrance road and some year I am going to manage to be there on a Monday so I can go on the ranger hike!
BTW, just because you have an itinerary doesn't mean you have to stick to it if something more interesting comes up (mama bear and cubs, wolves, pikas (lol), awesome geyser eruption, etc). It's just a guide to head you in a direction from your lodging with an indication of what you might see for the day.