Windows Or Mac?
If its a windows system you should consider the following advice (if you don't want to spend alot).
1. Get all Service Packs and uptodate patches
2. AVG Free is a great updated Virus Scanner, and the price is good. http://free.grisoft.com/doc/1
3. Depending on how savvy you are with Internet protocols and what not you have a few options for personal firewalls. Beginners should go with ZoneAlarm. Its pretty quick and easy to learn. If you feel you can handle a challenge then Tiny Firewall is your best bet. http://www.iopus.com/guides/freefirewalls.htm
4. Spybot is the best damn Spyware removal software on the planet. Also helps protect your Registry from being overwritten or modified by programs you install. http://www.safer-networking.org/
5. Get yourself a copy of the Windows Defender Beta from Microsoft. Scans and neutralizes malware as well as potential security threats. http://www.microsoft.com/athome/security/spyware/software/default.mspx
6. If you are also interested in seeing how secure your system is, you can also get the Microsoft Baseline Security Analyzer as well. http://www.microsoft.com/technet/security/tools/mbsahome.mspx
7. Download Firefox
8. Finally be sure to run applications that are designed to work with XP SP2. A lot of problems and headaches can be caused by installing older versions of software. For example installing an old version of Roxio CD Burning software causes problems with the system (Status of CD-Rom drive never changes, instability, annoying warning popups at start up.)
9. For the love of God don't install AIM, use an open varient like GAIM. Its less flashy but doesn't install that fucking AOL Browser on your system that you can't remove, as well as that piece of shit Viewpoint.
Install Linux instead or use VMware and browse the internet that way.
Edit: Oh didn't see your WiFi router post. The answer is yes. One, on most home routers the firewall sucks. Two, the router firewall won't protect you from crap already inside the network. Three, ZoneAlarm will also tip you off if some program or process is trying to access the Net without permission, which could be a Trojan trying to call home.