My preference for that type of excursion has always been Sedona, AZ. There are more than enough spectacular hikes in Sedona itself to go several weeks without repeating any, and there is surprising variety experience wise. Flagstaff can be added for even more options. There is an abundance of solid eats, many with spectacular views of the red rocks.
It's about a 2hr drive from the PHX airport, or 30ish minutes from Flagstaff. There are no direct flights to Flagstaff from Austin, so I usually fly to Phoenix, and roll in just in time for lunch. Vacation rentals are abundant for most budgets, and may even out number permanent residences at the moment.
If you choose Sedona, it can be hot in the middle of Summer. October is ideal, weather wise.
Another favorite is Telluride, CO, though it takes almost a full day of travel to get there unless you happen to already live in the Rockies.