Forming a contract
Forming a contract is easier to do and less formal than it sounds. Contract law (like most areas of law) has over the years attempted to codify the steps that people have gone through to make agreements. Every time you buy something from a shop or take a cab ride you have formed a contract. If I make an offer to my client and they accept or if they make me an offer and I accept, then we have formed a contract.
Traditionally we tend to confuse the contract with the piece of paper describing it. But in law, a written record is not necessary for a contract to exist, and a signed document does not in itself prove that any contract exists. Written records do provide substantial evidence should it come to a dispute. But written records are more useful in negotiations because it's all too easy to forget the details of the agreements we have made.
Offer and counter offer
To form a contract one of us must make an offer and the other must accept without conditions. The courts consider a counter offer to replace the original offer, and so we lose any terms in our original offer, but not in the counter offer. Therefore, if we wish to make a counter offer we should ensure that all of the original terms and conditions are copied into the new contract.
It is always useful to get legal advice before dealing in any legal matters, but there's no requirement that a lawyer must be involved in any of the contracts we tend to deal with. Legal advice can be expensive, so if the legal work would cost a large fraction of the agreement it may not be worth it.