If you're asking whether you need a lawyer to draft a contract, legally, the answer is no. Anyone can draft a contract on their own and as long as the elements above are included and both parties are legally competent and consent to the agreement, it is generally lawful.
Ing to Boundy (2012), typically, a written contract will include: Date of agreement. Names of parties to the agreement. Preliminary clauses. Defined terms. Main contract clauses. Schedules/appendices and signature provisions (para. 5).
In most cases, a contract does not have to be notarized since the signed contract itself is enforceable and legally binding in state or federal courts. Many types of written contracts don't require a notary public to be valid.