It's a huge area and I would try to break it into sections and then adjust itinerary once you are there and get a feel for place. I am consistently underwhelmed with food options in park, I'd plan on a cooler and brining your own food, picnic along way, multiple spots with great views. Consider buying some portable chairs in Bozeman.
Look at NPS map and you will see park is mostly a figure eight. You will want to hit highlights.
I would plan one day for old faithful and geysers, try and go early or later in day. There is a nice boardwalk system. From your local that will take a full day. On that day knockout all grand prismatic (there is a trail to get to elevation and see from above, I don't know if they've made it an official trail yet, it's been off the books for a while).
I would do another day, get up early and hit breakfast as Rosevelt (it's not bad), then south toward canyon, could do mt washburn hike (if you do there is a trail head as you approach from north, it's up a fire road and I think knocks down some distance. From there do the canyon waterfall look outs and the brink of lower falls, its unique. Head south, do your picnic along river shore, then drive into Haden valley to see some buffalo.
I really like the stretch from canyon through Haden and down around the lake but doing lake and all above is too much to get you back to camp.
You might consider a day to get down and hang out at lake, elephant back is good hike there. Take leisure drive back, cross over from canyon to Norris and back north.
Another option is across Lamar and through beartooth mentioned before, beartooth is pretty epic, you don't have to do it all, but go at least to ridge for views, you can scope it on a map. Or you could go to red lodge and then back to camp via interstate so you don't double back (big day).
I really feel based on what you are looking for you might consider splitting time between Teton and YNP. I feel Tetons have better hiking and mountain views, and YNP has natural wonders, wildlife, expansive valleys.
Look at park/vendor offerings, there is a chuck wagon dinner in Rosevelt, some motor coach tours. I'd consider a coach from north to geysers, let them do driving and parking. Theses are usually most crowded. There are also vintage yellow coaches that are open air and can be fun. Vendor offerings in park fill up fast, I'd book early if anything jumps out.
Would not recommend night driving in park too much, a lot of wild life.
It's bear country, keep that in mind and plan accordingly.
Enjoy, it's hard to mess up.
