It's good that you're thinking simple -- Fido probably doesn't want a lot of whereases and wherefores following him through life. But a basic bill of sale would be a good idea. It will help to ensure that you and the seller really understand what you're agreeing to, and will protect you if any of the seller's promises turn out to be false.
Nolo's book Every Dog's Legal Guide: A Must-Have Book for Your Owner has a sample, and it's separately available as an eForm. Or, you can start with a standard form bill of sale, then adapt it to include the terms you consider important in this transaction. For example, if the previous owners are making any promises about the dog's health, vaccinations, training, or pedigree, include these in the contract.