A firewall is pretty close to necessary, yeah. If you are set up like this...
-- [Internet] ------ [Modem] ----- [Computer]
then you should have a software firewall on the computer.
If you are set up like this...
-- [Internet] ------ [Modem] ----- [Router] ----- [Computer]
then the computer is protected by the router and a software firewall is optional but possibly still useful. If you have multiple computers sharing the router, then software firewalls protect each computer from the other computer(s) in the event that one of them becomes infected.
Only use one software firewall at a time so they don't clash or get confsued. What I would do, if you have enough Ethernet cables (which maybe you don't), is to get your router in place, update your antivirus software, and run a thorough antivirus scan.