We have stayed at Midtown Residence Inn not the one on 17th Street. It is about a city block to MARTA which you then take to 5 points station. There is a Braves shuttle bus that leaves from Underground Atlanta, adjacent to the the MARTA station. You walk through Underground Atlanta to get to the free shuttle. Buses are lined up and leave as soon as they fill and drop you right in front of stadium... Well laid out. The return is easy as well.
The hotel is across street from a good restaurant with a great crab dip and is just a couple of blocks to a small grocery store, drug store and liquor store. The rooms at this Residence are quite large by the way. It is a great place to set up for a few days and catch some games. We are headed back in August.
If you're flying in, you can stay at the ATL Airport Marriott (Convention Center) or the adjacent SpringHill. They're next to the tram into the airport terminal, and from there is the MARTA. Had a couple coworkers do exactly that a few weeks ago. The tram is free, the MARTA's only a couple bucks, and the game, well, you couldn't pay me to watch the Braves.