No advice on flying...since I usually drive, lol!
Here is a suggested itinerary.
Day 1: Arrive Bozeman. Stop at WalMart or another store and pick up a cooler, lunch fixings and beverages. Depending on what time you arrive, if it is around noon (your stomachs will still be on Eastern time so you may be hungry anyway! lol), go ahead and eat lunch in Bozeman. Head south on 191 To West Yellowstone and enter Yellowstone thru the West gate. It's 1.5 to 2 hours from Bozeman to West Yellowstone. It's about an hour of drive time from the West gate in to Old Faithful with no stops, however, plan on a lot more since there is a lot to see. You will likely see your first bison and elk on the west entrance road, or in fact on 191 as you head down to West. Take your time on the West entrance road, then at Madison Jct turn right. Madison has flush toilets and makes a good pit stop. After you leave Madison in the direction of Old Faithful you cross the Gibbon River Bridge. Take the next right into Firehole Canyon Dr. Re-enter the Loop Road and continue south. I would probably head straight to Old Faithful area skipping the geyser basins you pass as you can pick them up tomorrow. Arrive at Old Faithful, check in at the Visitor Center, pick up the brochure at the desk for 50 cents and write down eruption times of the predicted geyers if any are predicted for early evening. Depending on time of day, get ice cream on the first floor of the Old Faithful Inn and take it up to the 2nd floor deck and enjoy eating it while overlooking the geyser basin. Overnight Old Faithful
Day 2: I would start back at the Visitor Center. There is usually a Ranger-led tour at 830 for about an hour and a half and it is usually quite good. If there are any prediction times up, note them down. Walk thru the Upper Geyser basin. If I am not on a tour, I usually start by walking around to the left of Old Faithful, going behind it and crossing the bridge over to geyser hill. Work your way down to Morning glory pool on the Boardwalk area until it meets the paved path. There is a vault toilet at Morning glory. Head back to the Visitor Center/OFI on the paved path. I like to eat lunch at the Lower General Store. It has gas pumps out front and is the first building you come to when you are headed back from Morning glory. Hop back in your car and head back toward Madison Junction picking up Midway geyser basin, Firehole Lake Dr (check for time on Great Fountain eruption and if there are people waiting stop. It's magnificent!) You exit Firehole Lake dr right across from Fountain Paint Pots, but you can't go straight across, so turn left or right and then hand a u-turn in the next pull out. Depending on how your time is running, you might have time to head to take a look at Gibbon Falls and Artist Paint Pots which are on the road between Madison and Norris. I doubt you will have time for Norris today. Return to Old Faithful. See what the Ranger program is for the evening. Overnight Old Faithful
Day 3: Head to Grand Teton. I would probably not stop much in Yellowstone. Skip West Thumb geyser basin for now. Overnight Jackson
Day 4: On your way back in to Yellowstone stop at West Thumb for a quick walk thru. What you see today will also depend on how much time you spend in Grand Teton. If you want a short hike today some choices would be Natural Bridge, just before you get to Bridge Bay or Storm Point which is east on the East entrance road from Fishing Bridge. Drive thru Hayden Valley looking for wildlife or for people with large camera lenses or spotting scopes. If you have a chance, stop at the Visitor Center at Canyon. Overnight Canyon
Day 5: Often people will take a drive down thru Hayden Valley to look for wildlife before breakfast, then either return to Canyon for Breakfast or go on to Fishing Bridge. If you begin from Canyon after breakfast, be aware there is a rainbow over the Lower Falls at 945 from the Artist Point overlook on the South rim. This one is an easy, level walk out to the overlook. You might also want to do a ranger led hike this morning. Do the overlooks on the South and North rim. Head south thru Hayden again stopping at Mud Volcano and then Fishing Bridge. Park and walk across. You are not actually looking at the bridge, you are looking for wildlife from this vantage point. Perhaps otters, golden eagle, white pelican, muskrat. Head back toward Canyon stopping at Le Hardy Rapids. If you are looking for a picnic spot in this area, I love Nez Perce Ford picnic area. It is back off the road and right on the river. Head north to Canyon. This is probably a full day especially if you have gotten in a hike. Overnight Canyon
Day 6 From Canyon, head north over Dunraven Pass. Stop at Tower Fall and overlooks north of there. At Tower Roosevelt Junction, turn left toward Lamar Valley. If you are looking for a hike, I like the Yellowstone Picnic area hike, the trailhead of which is just past the Yellowstone River bridge on the right. You will need bear spray. If you wanted to look at a hike up Mt Washburn this would be a good day to do it. You would need to check with the Rangers at Canyon VC to see if it is open as it can be late to get the snow gone up there. Drive out in to Lamar Valley and look for wildlife. Many like the Trout Lake hike in this area as well. If you don't want a hike but do want to drive Beartooth Highway, you could do it this day. Drive out the NE entrance and drive over Beartooth to Rock Creek Vista, returning the same way. It is OK to back track! Depending on how much time you have, you might be able to fit in the very short drive back to Petrified Tree (look for Black bear) an then the drive on Blacktail Plateau Drive which exits right by Petrified Tree. Head to Mammoth. You likely will not have time to do Mammoth this afternoon. If you need a pit stop, the best bathrooms are by the gas station and bus parking lot. Overnight Gardiner
Day 7 Re enter the North entrance. Stop at Mammoth and go to the Visitor Center the do the Lower Terraces. Drive to the Upper Terraces which will be on your right as you drive up out of Mammoth. Head south again. If you are near lunch time, I like Sheepeater Cliffs for a picnic area. Head down to Norris Geyser Basin and do Porcelain Basin and Back Basin. I like the small museum of the National Park Ranger which is by the entrance to the Norris Campground. If you didn't fit in Artist Paint Pots the other day, you can head toward Norris and see them now. You can also head over the connector road between Norris and Canyon and do the Virginia Cascade drive. Overnight Gardiner
The disclaimer on the itinerary is that it should go out the window if something interesting comes up like wolves, bear or a spectacular geyser eruption! Also if you don't like one area you don't have to stay! I usually spend a whole day in the area between Madison Junction and Old Faithful and another whole day between Canyon and Fishing Bridge.
You will need to buy bear spray when you are in Bozeman. You really should not hike without it. If you are out real early in the geyser basin around Old Faithful you should have it.
You also want to get to your lodging in the park before dark. That time of year you will have lots of daylight, so shouldn't be a problem.
You asked how reliable GPS is, and I am not sure what purpose you would use this for? There is really no need within the park as the road is a figure 8 and it's really straight forward to get from the airport into Bozeman and on to 191 and back from Gardiner to I90 and return to the airport. If it is handheld, someone who geocaches recently posted about finding Earth Caches in Yellowstone.
Also many of us like Janet Chapple's Yellowstone Treasures guidebook which gives a lot of information.