OP, what time frame are you working with?
My suggestion for a "no destination" road trip is to buy a national parks pass, then spec out a series of national parks that are kind of in a row, and visit them; linking them together with random non-interstate roads. This is a good way to sort of stumble across the country and see a lot, without giving you a solid "end" destination. I find that having a destination locked in kind of ruins a trip for me, because then I always think I'm on some sort of timetable to get there. Vague meandering is best. Plus you got to keep it loose, you never know when a road / park is going to be shut down... I've been unable to go through Glacier TWICE now due to freak snowstorms...
My wife and I did a 13 day trip from Ohio to California and back in late 2015 doing it this way and we had a great time - epic parks, lonely off road, oceans, deserts, mountains... but be careful... we ended up moving to California because of our trip!
In our case, we slabbed it overnight to the Badlands, then did Mount Rushmore and associated roads in the Black Hills, Yellowstone, Tetons, Crater Lake, Redwoods, Pacific Coast, Death Valley, Grand Canyon, Moab, Rocky Mountain trails (Engineer, Cinnamon, Imogene)....
On other trips I've done a lot of the stuff suggested above, and I concur... RMNP via US 34 is awesome, rural Arkansas and Texas Hill country is great, US 30 in the Columbia River Gorge...pretty much anything in Colorado, Montana, and Wyoming is going to be incredible... really, it's hard to go wrong heading west...
If it gives you any ideas, below is a link to a the trip we took. Ignore the Xterra... that was before I came to my senses (JK, actually, it was / is a great truck!):
https://meefzah.smugmug.com/Sept-2015-4WD-US-trip/
What the hell, as long as I'm pimping out my shit, here's our move out west following the taking of that cross country trip...
https://meefzah.smugmug.com/California-Schemin/
Anyway, even if you don't move as a result of an epic journey, you will find your horizons broadened, you'll meet new people, and you'll renew closeness with your family (well, with 2 kids along, you actually might do homicide or two along the way, but here's to hoping they are trip-friendly...). At any rate, assuming no homicides, you will find that the world is simultaneously much bigger and much smaller than you could have ever imagined.