Befriending my enemy??

So last year this girl in my grade and I had a really bad relationship over stupid things. I started to hate her and hold grudges because she did some things to my friend that were pretty unnecessary, but everyone makes mistakes and I’ve moved on. I bitched about her behind her back, which I deeply regret. Afterwards I just avoided her as much as possible and this went on for a couple months. So after this period of time I heard she was talking about me behind my back and how much she hated me, etc. I was a little confused because I would’ve expected it back when I was being a bitch about her, but it just sort of came up out of the blue. It’s irrelevant anyways. I just really wanna be a better, kinder person, so I think a good way to start school this year would be to apologize to her and ask if we can start fresh. I’m already a very socially awkward and anxious person, but I know I need to do this. Do you have any tips on how to approach her and show my sincerity? Any advice will do.

(Plus if anyone has suggestions on how to improve myself as a person in general, I would really appreciate it.)