Congratulations, and welcome to the SF forum!
Jane has given excellent advice, so I will only add a bit.
I suspect that what your immobilier is drawing up for you to sign in the first instance will be the offre d’achat.
The immobiliers tend to use a template for this. Your requirement to include the Certificat d’Urbanisme clause should be included within this offer. It is best that the seller and you are both on the same page from the start.
Next, either a promesse de vente or the compromis de vente
I would highly recommend you request the compromis and have this drawn up by a Notaire. Please note the caveat outlined regarding using the immobilier for this.
Contracts for property are serious commitments in France and should be carefully considered. The Notaire is effectively being paid by you, so it would be good to have one you can trust.
May also be worth reading:
The ‘cooling off’ 10 days mentioned by Jane:
“A deposit of up to 10% of the purchase price is made on signing of the agreement (compromis de vente) and the purchaser has 10 days during which time they can withdraw from the contract without penalty.” Deposit will be held by the Notaire. (And returned directly to you if you withdraw before the 10 days lapse.)
Then you wait for the Notaire to check all that must be done and takes ~ 3 months
All being well, the acte de vente drawn up by the Notaire will be signed by both yourself and the seller, with balance of payment for the house transferred by you to the Notaire, who then divides taxes owed between you.
About 6 months after the acte de vente signing between yourself and the seller, the Notaire will send you your deed but the house will be yours from the day of the signing.
Then the real fun begins!