2 Points of travel I recommend are at the very end, and in new mexico. Houston up until new mexico you'll pretty much be on paved roads. NM it starts to get "Dirty"
The first one goes through the Valle Vidal of the Carson National Forest. I've posted about a fun little easy trail back there, but this is pretty much the loop
Second one is from the North Kaibab Ranger office to the north rim. a bit on pavement, but a fair bit on really bad dirt roads/2 tracks.
Your best bit is to stay in national forests the duration of your trip, as that's where you'll hit the most "reliable" dirt, If you plan to do it over 2 weeks. I spent 11 days driving from Dallas to Mammoth Lakes, and I was able to do the two trails, but not a whole lot else. that was 3,000 miles tho.
If you want to have a fun day trip, do the 4x4 trail at the great sand dunes national park. worth it.