I got married nearly twenty-five years ago, (my wife is Catholic, I am an aetheist - it's best not to get into that discussion with the priest - I didn't!) and the Catholic church has eased its restrctions in that time.
As advised by previous posters, I had to confirm that any children would be raised as Catholics We had the three talks about marriage, which was great becuase our priest was Father Tolkien, son of JRR, and his study was full of artefacts from the great man's study in Oxford.
I did have to prduce my baptism certificate to prove I had been baptised - not sure if they still insist on that now.
Best thing, as advised, is for both of you to start going to Mass at the church in which you want to marry. If that is not your own parish church, the priest will need to get permission from your parish priest - this is usually granted, it's just a courtesy between parishes.
Have a chat with your priest, and he will advise you of all the proceedures needed.
It is a tradition to offer the priest a gift on the day - he probably won't mention that - an envelope with some folding will go down very well.