Welcome to the board, Scott.
Your vet will keep copies of the records, and will give you the originals at the time of service. Or vice versa, depending on the vet. Either way, you'll get a copy for yourself.
I put mine in a manila folder with the horse's name on the tab. That way, I know who the vet has seen and what's been done for each animal.
You can keep the horse's negative coggins in this folder too, and make copies whenever you want to take the animal someplace that requires a copy of it.
As far as wormings, that'll be up to you. I worm my horses on a rotational, 8-week schedule. There are worming charts you can find on the internet that will take the guesswork out of it, or you can have your vet recommend a particular routine.