I am a practical guy, and I have two sons (grown now). Taking trips when they were young, they were good in the car for about 3 hours (4 max), and that was with games, music, stories, etc. So I would not try YNP at this time in your family's life. YNP is over 26 hours away (drive time). I would save the western trip until your kids can fish with you and truly enjoy it.
I agree with aiming up the east Coast. State College, PA is just over 15 hours away, and the Farmington River, CT is @ 19 hours. Going in that direction covers a lot of rivers, with many at the destination too (if you get that far). Plus as others have said, there is American heritage and history, as well as exposure to cultural diversity, as you head into territory that is politically different and which also provides new food experiences. Up the East Coast the population is denser so there is good shopping as there are more attractions per mile (not everyone fishes).
Good luck whatever you choose.