I think every place is close to a gym. I live within easy walking distance of 5 gyms. Amazon Gym in center is small but nice. Probably the nicest gym is at Hermes hotel on Savanoriu near the highway. The most serious athletes can probably be found at the sports complex near the zoo - lots of university athletic majors there.
Every apartment is also close to a grocery store. You don't have to think about such things here. It's not America. 
I guess you've probably already found a place to live. If not, you're in trouble, unless you go expensive, because the students have snapped up everything cheap this time of year. But anyway, if you're worried about traffic noise, then you should also be worried about drunk noise. Laisves al. can be noisy in that regard. The old town (senamiestis) is quieter and a little friendlier to foreigners (more English speaking waiters/waitresses, more tourists.) If you live in a courtyard off Vilniaus g. (a pedestrian street like Laisves), it will probably be very quiet.
Drop be a message if you or anyone else want to meet for a drink at a cafe.