Once you find a good playgroup/mums & tots group you will meet other mums & dads & build up friendships. Find one that accepts babies & start as soon as you like. Then, when she is older, you can have playdates. She will be fine, just means that you will have to be her play buddy some times!
But it's also important that she learns to amuse herself too. But that's a while off yet!
I'm sorry you suffered during the birth, but please remember that the most important thing is that she is happy & healthy now. You won't forget all about the birth, but in time you will come to terms with it. I used to cry because I didn't hear dd crying when she was born as I was still under ga. (don't even know is she did cry, tbh) I cried that DH didn't get to cut the cord, I cried that I had to go thru it all alone & leave DH in the corridor and that my last memory of him that day was him with a tear in his eye, as he felt so helpless, that I wasn't the first to dress her/hold her/feed her. But in the grand scheme of things those things are not so important anymore.
I do understand that these things are different from feeling pain tho and I do sympathise with you. I hope to have another, but am terrified of the thought of giving birth either naturally or thru elective c-section. But I know if I want another child I will have to do it.
Oh, and the tiredness will soon fade too. Once your little one is sleeping thru etc. Of course, it's a good excuse for a nap for you too..(I still do if dd has had a bad night and goes back to bed, one of the advantages of being a SAHM!!