We got home from Yellowstone and Grand Teton about week ago. A couple of suggestions to consider for your route:
We like I80 across PA, but then use I76, I71 and route 30 across OH and IN to I65 to rejoin I80 near Gary. It's about 20 miles longer but no tolls, much less traffic and few trucks. Route 30 is a really nice divided highway with some traffic lights toward the western end before I65.
We also continue west on I80 to I29 then north to I90 at Sioux Falls. I35 between I80 and I90 is another option.
Since you have an extra day on the way home, I'll throw out another suggestion - Take route 26 out of Grand Teton to Riverton then route 20 to I25 at Casper to route 26. Route 26 follows the North Platte River and a portion of the Oregon Trail. At Guernsey WY are the best preserved wagon ruts cut about 4 feet deep into a rock formation. Two Miles away is Register Cliff with names of immigrants dating back to the 1800's still visible. A few miles further on 26 is Fort Laramie, with a number of restored buildings. Further along on 26 you will pass Oregon Trail landmarks including Scotts Bluff and Chimney Rock before rejoining I80. At North Platte NE is the Bailey Yard of the Union Pacific RR, the largest rail yard in the world. The Golden Spike Tower there affords an elevated view of the operations.
Enjoy your trip!