Look at all you fancy international travelers. I'm a bit of domestic, gypsy, road traveler myself. I've driven everywhere through the Western half of America except for Washington State and only a decent bit on the Eastern side. Doing these trips solo every year is kind of my thing, while I reserve the beach and snow skiing for group trips.
Hands down, the most adventurous and amazing trip I've taken was the off beaten path through the Grand Canyon. I hiked about 40 miles all in all and went down to Havasupai and camped by the blue waterfalls. It wasn't easy to get there, but totally worth it in the end. Here lately it's become a little more publicized and known and I'm not sure how I feel about that, but absolutely do plan on going back someday.
Outside of Havasupai, one of my biggest recommendations is taking the 1 from LA to Washington State. The Californian and Western coastline has some of the most remarkable scenery I would imagine on the entire planet. That was a pretty special trip. Be sure to use your map and plan ahead. What I did was hugged the coastline on every road I could, the entire way up. Staying on just the 101 is way too easy. The bad news though is you'll have to wait until the road gets cleared up in Big Sur. That landslide has taken well over a year to fix and it still isn't done.