The great thing about Austin is that there are many neighborhoods that fit your criteria, the bad news it that there are many neighborhoods that fit your criteria. But none have it all.
Staying close to downtown AND finding a neighborhood with playgrounds, pools and tennis courts will be the challenge. Many of the the newer neighborhoods have these amenities, but those close to downtown don't. While there are a few pools and parks in the older neighborhoods, the most popular place to visit is Zilker Park and Barton Springs or Deep Eddy pools (both natural springs).
The biggest variable will be the schools. Many nightborhoods have elementary's close by, but middle schools may not be as convenient, if that's important. Performance is also a wide gap. Traditionally, Eanes ISD is excellent, and there are home in Eanes within 10-20 minutes of downtown. Another cosistant district is Lake Travis, although you will lose the closeness to downtown. Both these areas are very nice, but just a few of the neighborhoods have the community amenities you are wanting (Senna Hills, Barton Creek West, Lake Point to name a few). If you go further south there are some neighborhoods with the amenities, but you are leaving downtown (30-40 minutes) Circle C, Baurele Ranch and many more. You can also go north to Steiner Ranch, RIver Place, Avery Ranch and a host of other fabulous planned communities, but the commute to downtown is much further. Check out this link for more school information http://www.alamotitle-austin.com/library/23/Public_and_Priva
I would be happy to help you with your move, if you would like to visit please either call or email and we can get started.