Congrats on the gig, that is always exciting.
As you are trying to keep it professional, I would most certainly write up a contract. This could be as simple or detailed as you want to make it. Likely the format of your contract will change as you learn from each job.
Some pros for the contract, other than the professionalism, is that it will spell out the price (even if it is already agreed upon) and the specifications of the site. Any developer can tell you scope creep can eat into profits very quickly.
As you know the client, simply tell them that the contract is you learning to do things professionally. If they are not looking to get one over on you they should have no issues with signing one.
Good luck and have fun!