Arlington has a lot outdoors, but it may cost you a bit more. You can see a Rangers game, visit the Cowboys stadium and go to Six Flags and/or Hurricane Harbor. They are all walkable from each other.
I echo the above on public transportation. Dallas is much better on that front...and it still isn't great. I'd take an express bus to New Braunfels and tube the river on a weekend. The Hill Country is a much better option for outdoor activities.
And yes, it will be very, very hot.