Current UT grad student:
As far as living in Austin, there are many places grad students live. Hyde Park which is about 1.5-2 miles north of campus is a popular neighborhood for grad students with easy bus transit to campus and the city. It has a mixture of affordable apartments and houses with families. Far West is another area I've been told is popular for housing, but it's further removed from town and campus. East Austin is a very growing, but gentrifying, artsy neighborhood with new residences, but costs are going up there. West Campus, where undergrads tend to live is probably a place to avoid as it's particularly expensive and noisier.
One of the best first steps is to contact an apartment/property finder here, there are a lot of them to choose from. Providing them your needs (furnished; bedrooms/efficiency, etc) and price range, they can find a few suitable locations with no commitment (they get paid through leasing office if you sign). Also have known students finding housing and roommates through Craigslist, but that requires a little more effort and forethought. While one can find affordable single-person apartments, the majority of people elect to have a roommate to split costs with and rent come out a little under living alone.
There's a lot to see and do in Austin; there are graduate student organizations, but not too many. Met other students primarily through just going to campus events and through cohort and also volunteering in the city.
Hope this helps answer some of you questions.