Wow! What a nightmare three and a half years you have had, and well done for sticking to it and doing the right thing. The way you have handled this, and admitted your mistakes, is admirable.
Do you have contact with any other children at all? Friend's children/siblings, kids in your neighbourhood for example? Or are you having to learn how to deal with little people from scratch?!
My advice would be to go for the first meeting with a pretty box, one that you've either made yourself or one you can buy from a card shop to put gifts in. Tell her it is her special box for her to keep things in that you will send her in the post each week. You can choose together what she would like to collect, it may be postcards, fridge magnets, a comic, Mr Men or Little Miss books, but something small and light to post, cheap for you to buy each week and something that you can reliably send each week. Although she knows you are her daddy, you are a bit of a stranger to her and she may be wary of having gifts from strangers, and she may expect gifts off you every time, but this way, she can look forward to a little something that will arrive through the door and she can treasure it in her keepsake box. That may sound odd, but you don't want her to like you because of what you bring, you want her to respect you and love you because of who you are. Parents do not ALWAYS bring their children presents, sometimes they have to do with our presence instead! So you can send her gifts when you are not with her so that she knows you are thinking of her, then she can look forward to the visits to spend TIME with you.
If you do see her grand mother, I would suggest a courteous 'hello' and a smile and that's it. If the lady ignores you, that's her issue, don't make it yours. My ex hated my mother and would totally ignore her or insult her and this did upset my son. Remain the mature sensible one, and if she responds negatively to your greeting then take it in your stride and continue on your way.
I'm sure the contact centre will have some games or toys to play with, so ask her what she'd like to do and go along with it. There may be a jigsaw you can work on, or a story you can read her. Answer the questions as simply and honestly as possible about why you haven't seen her for all this time, something like "I would have loved to have seen you but I didn't know where you were, but I've thought about you every day and knew that one day I would be able to do things with you" may satisfy her. Avoid things like "I wanted to but your mummy wouldn't let me" or "mummy has not been well" or "I've had to take mummy to court so I can see you". These are things that she will learn over time and have to learn how to deal with when she's in her teens (or perhaps younger). Try to be as economic with the truth as you can, but don't tell lies either.
When it comes to physical contact, let her be the one who comes to you for affection, hugs, kisses in her own time. She may not feel comfortable hugging you or kissing you on the first few meetings, so don't be upset if she doesn't.
Talk to her about her school life but without asking direct questions of who, where, why, what etc. Find out what her favourite lesson is, who her favourite teacher is and why. Ask her what makes her laugh, who is her funniest friend, but by saying something like "tell me about a typical day at school", "tell me a funny thing that's happened with your friends that made you laugh", "I bet you have a favourite subject/teacher/lesson" and see what she responds with. Too many direct questions may sound like an interrogation!
Your time will be over before you know it, so when you leave, remind her of the special box and let her know that you will choose something and post it on a certain day so it should arrive by a certain day.
Hope this helps a bit. Good luck.