YES you need a contract.
It should include, at minimum:
- What your deliverables are.
- Where you are expected to work / who provides equipment and software.
- Who to invoice / when you will get paid.
- Assignment of legal costs to the prevailing party if legal action has to be taken to collect.
- Interest rate / fees for late payments.
- Start / End date of contract.
- Notice period for early termination for both parties.
- Who you report to.
Also, you should figure your rate at about DOUBLE your salary (my opinion). In the U.S., you will now have to pay your own insurance (wait until you see what COBRA costs), additional taxes, and "savings" for the down time expected at the end of your contract. If the contract is 6 months to a year, maybe a little lower.
See if you can find someone in your field who is contracting in your market, and ask them for advice. They may even have a contract you can use as a starting point.
You may not need a lawyer, but make sure that everybody's expectations are in writing. Nothing is just "understood" when it comes to contracting.
Thoroughly document your deliverables as you complete them, and do it on your systems, not theirs, or at least BCC your personal email with them.