You need a bill of sale. It needs to outline how much you are paying, that you have paid in full (pay with check or money order or something traceable, for your own protection).
Check on whether the horse's vaccinations, worming and coggins are current - if not, you'll need to schedule an appt with the vet ASAP.
Get the number for the vet, farrier, etc that the seller has been using. Talk to the Barn Owner of the facility in person and review their boarding contract to make sure you understand the terms and that they work for you.
You need the papers, if the horse has them - who the horse is registered to isn't all that relevant, as long as the necessary transfer paperwork has been signed. If it hasn't been signed, call the registry before paying for the horse and check what will need to be done in order to get the horse transferred into your name (you may be able to get the seller to lower the price for the horse by whatever extra it will cost you to get him properly registered if things are not in order).
As for the vet check - by all means, get one. However, realize that the results are no guarantee of anything other than the horse appeared healthy and sound on the specific day and time. The basic vet check for a non-competition pleasure horse is basically the equivalent of kicking the tires on a used car and driving it around the block.
The vet will check that vaccinations, coggins, worming are up to date - and probably perform a flexion test to check for lameness.
The thing is - some relatively healthy horses may appear lame after a flexion test due to arthritis or other conditions that may not have any real relevance on what you will be doing with the horse. My own guy has a chip in one hock and is occasionally mildly off in that leg - however, it hasn't affected his usefulness or performance and is fine with regular care and maintenance. He'd probably fail a vet check on the days when its acting up - but the other 90% of the time you'll never be able to tell - to the point where the BO at the boarding barn called me one recent (chilly) morning (he's been there several months) and mentioned that she was calling because he seemed off in his right hind - I realized I had forgotten to mention the bone chip to her. She'd seen no sign of lameness in the 3 months prior to the overnight cold snap.
The point of this little story being - unless you have an EXTENSIVE vet test done - complete with xrays, etc, like you would on a top level performance horse - there is a good chance the vet won't detect an underlying problem anyway. Furthermore, many problems can be hidden by an unscrupulous owner with a couple tablets of bute or a tranquilizer. The vet can not always tell if a horse has been drugged, especially if the drugging is subtle.
Should you get a vet check? It won't hurt anything - but you'll probably get just as good an idea of the horse's health by talking with the owner, the BO and coming out to the barn a couple of random times just to see the horse in action.