H.C.
I was in the same situation about 2 months ago! You might search for the responses I got. I think the title was Austin neighborhoods or something like that.
Brentwood/Crestview is about 5 or 6 miles from downtown, with a park/pool (Brentwood Park).
Allendale is 6 or 7 miles from downtown, with a park/pool (Northwest Park).
Travis Heights is super-close to downtown, 2 or 3 miles (Big/Little Stacy Pool).
That's all I can think of right now. If you don't get any other responses, PM me and I can tell you more. Welcome in advance!
H.