Voice and text, of course. It has a Band 17 LTE radio, so it'll do data too. It has to be unlocked to take an AT&T SIM. If it's branded Google, it's unlocked. If it's branded TMobile, you'll have to read T-Mobile SIM Unlocking.
It absolutely should work on AT&T... T-Mo does not sell locked phones so your N6 should be fine on Big Blue. Walk it into an AT&T store and have them test it with a new sim if you're not sure.
However, if T-Mo's service is bad just at one location, say, your home, ask them to help you out. I'm on T-Mo, and my service is great. But my father-in-law signed up with them and service (from ANY carrier) at his home is lousy. He told T-Mo that he wanted to return his phone and cancel service and they provided him with a device that amplifies the signal in his house... it still isn't great, but significantly better than before.
T-Mo's customer service is materially better than AT&T, and certainly less expensive. You know that you'll have to pay off the balance on your N6 if you part company with them, right? Hope this all works out for you.
EDITED: I stand corrected... I read RUKBAT's post and followed the link to the T-Mo unlocking site... I've been told by them that they do not sell locked phones, but this link contradicts that. It would appear that you will have to pay your phone off first, then get an unlock code from T-Mobile before using it on AT&T's network.