Here's a route you could take if you were going up from I-80.
http://wyomingnaturalist.com/html/dirt/sec_7_9/S8_T4.html
And here is the link for the entire online version of Wyoming Backroads Guide
http://wyomingnaturalist.com/html/dirt/dirt_book.html
I lived in Etna (in Star Valley, just south of Alpine) from 2006-2015 and drove the first route a bunch of times for work as a Sheriff's Deputy doing forest patrol in the summer (most all of it is within Lincoln County) and with the family on weekend outings. It runs from Labarge in the south to Alpine in the north through the Bridger-Teton National Forest. From Alpine, you then get back on the pavement and follow US 26/89 up Snake River Canyon to Hoback Jct then left to Jackson and then through to the south entrance of Yellowstone via Grand Teton NP.
I didn't see what time of year you were planning to go, but if you're planning to take as much dirt as possible, I wouldn't suggest trying any of it until at least late June. There are places through there at altitude that may still be holding snow in mid-July (or later depending on how bad the winter was), so make sure you contact the Forest Service offices in the areas you plan to go to check road conditions and closures.
My avatar pic was taken in Grand Teton, along a two track that I can't remember the name of for the life of me right now, the year before I had to move kicking and screaming back to Louisiana. Man, I miss that area....
Hope the links give you some ideas!