Re the concept of a "day trip" to Arlington from Ft. Worth -- used to be that those without a car could take a Greyhound bus between the two cities, then walk about a mile to the stadium from the Arlington station. However, it looks like this service is no longer offered -- but back when it was, planning to spend the better part of the day at the game and in transit would surely have made this a "day trip."

I think you would have the most fun in the Austin/San Antonio/Hill Country area, with maybe a trip down to Corpus Christi or Galveston if you want to see a little of the gulf coast. Galveston is a cool town with a lot of history, although it was severely damaged by a hurricane a few years ago. Corpus Christi is nice enough, but I like the "vibes" in Galveston better.

You have received great advice above.