Since you already know what car you want, you should start by applying for pre-approval at multiple lenders. In a 14-day period, you can apply to as many lenders as you want and it will only count as a single hard inquiry against your credit score. This can help you avoid giant drops in your credit score while you search for a car loan.
At the closing, the lender will do all the paperwork, especially if the seller has a lien or loan on the car. In this case, your lender will make the check out to the lienholder rather than the individual, which all but ensures that the seller uses the money to pay off the loan rather than taking the money and running.