I wouldn't advise bringing bikes for YNP. The roads don't always have shoulders and the traffic is heavy. The Tetons have a bike trail in some areas.
Our bikes never got off the racks when we were there. All of the things in Teton and Yellowstone are a good distance apart so you definitely did not ride from location to location. A little tootling around in the campgrounds might have been OK, but it did not really happen.
If you are looking for the nastiest, boringest route it is I-80 to I-5 straight into Portland. Shorter and not too hard is 395 to Susanville to the 49 to the 89 and up the i-5 from Shasta. That is likely the route your GPS is using and is a good road with no big climbs. a Much nicer drive.
But probably to be considered is if you are going to do the whole drive in one long painful slog. If you are planning to stay somewhere on the way, I think the recommendations would change.