When i went to college there 20 yrs ago TT was a decent area from what i remember. I moved back again to Tampa about 7-8 yrs ago for about one year, and Temple Terrace has been going steadily down hill.
From what I saw, North Tampa (new Tampa), Carrollwood and futher north, and Wesley Chapel are much nicer areas, and of course the S.Tampa area Hyde Park and around there.
I live in Tampa and would be glad to provide some guidance. It truly comes down to what amenities you want and that will help your search.
South Tampa (south of Kennedy) has always been THE place to live in Tampa for the in-crowd. Great nightlife and a concentration of restaurants and bars.
Brandon (and neighboring bedroom communities) are nice, affordable but the traffic is bad.
Further south.....Apollo Beach, Gibsonton are nice places but a drive from Tampa.
Carrollwood has an older population but home values are stable and traffic is not as bad as New Tampa and Brandon. I like Westchase area the best and would look there is I were looking to move.