J.P.
My son started sleeping through the night at about 9 weeks old. Leading up to this, we would have a couple nights of full nights, a couple of waking up. She is learning, and you just need to be patient. You can try to let her go a little longer when she wakes up in the middle of the night to see if she can put herself down before you get there.
As for going to sleep, I used Babywise, and it worked great for me (my own modified version). And I started it at 3 weeks, but I know that others will argue that that is too young. Neither of my children have had an issue with sleeping through the night early with that method or learning to self soothe. Good luck.