Don't go in peak tourist season. Go either before Memorial Day (at which time there may still be snow) or after Labor Day.
IMHO the crowds combined with the distances you have to travel within Yellowstone, really take a lot away from the experience.
How much of this you take time to see on one trip depends on your future plans. If this may be your one and only trip that far out west, and you have essentially unlimite time, then yes, take time to see it all. If you expect to be able to go multiple times in the future, then concentrate on one area like Yellowstone.
If you want to avoid major cities for most of the route, take I-57, 64, 55, & 74 up through Illinois, and cross the Mississippi at Moline/Davenport. (Use IL SR4 to cut across from I-64 to I-55 without going into St. Louis.)
Then pick up I-80 across Iowa, I-29 up to I-90, and on across South Dakota, if you want to go to the Black Hills.
If you skip the Black Hills, stay on I-80 all the way to Cheyenne. Take I-25 up to Casper. Get off the interstates and go up to Thermopolis and then to Cody. This route lets you avoid pulling the trailer over the Big Horns.
Lots of good things to see on that route. Thermopolis is a neat town, with a big state park and hot springs. (No camping at the park.)
IMHO I would combine the Black Hills with Glacier National Park.
Find out when the big Sturgis Bike rally is, and avoid that week, plus a week on either side, unless you LIKE lots of motorcycle noise in the most inappropriate places.
Do NOT attempt to pull your trailer over Beartooth Pass.