The reason why I would prefer to stay at one place is the planning for a car and hotel. If we have to find a car and hotel every day, that's going to take more effort and take away from my training. We can hit all kinds of unfamiliar airports from one central place, we do it here. I've been to at least 10 different airports.
It's not difficult at all, in general. On my "grand adventure" I hardly stopped at any of the places I planned to stop overnight - I ended up just flying until shortly after sunset, pointing my airplane at the nearest small town with an airport beacon nearby, and going from there. I always managed to get a ride from the airport manager, or hotel, or a taxi that was a 1980's-era station wagon, etc... It really made the trip fun, discovering new places and meeting new people.
You really don't need a rental car every night - I never rented a car. Saved lots of money that way!
Also, it's not about just unfamiliar airports - It's about unfamiliar terrain, unfamiliar weather patterns, different controllers, etc. that you're not going to get if you're basing at a central field and flying back and forth out of there to a bunch of different, but nearby, airports.
However, if you're dead-set on "basing" somewhere, let me pile on with everyone who says to head to TX. Weather's good, people are generally friendly, and TX/OK have a number of spots with amazingly cheap fuel. Also, second the reco for Missouri on the way down and back, lots of spots there with cheap fuel as well.
As for where in TX - Stay away from the airports that are smack in the big cities, as they'll tend to be expensive (fuel, tiedown, and lodging). However, there are spots nearby that are significantly cheaper. For example, in the Houston area, Baytown. Not far from D/FW, Sherman or Granbury. I'm sure you'd do well if you went and stayed in Jay's hotel in Port Aransas, and I bet you'd have a good time if you went farther west (Marfa area) and hung around with Dave Taylor as well. Finally, I've heard really good things here on the board about Borger, TX.
Have fun!