There's nothing to stop you connecting without antivirus, but you'd have to be extremely careful about your browsing; and I'd suggest using Firefox with various extensions including noscript and adblock installed to stop malicious scripts running. Though having said that, antivirus won't stop a script installing a trojan, you need other software to do that.
There is a school of thought that says a combination of common sense and careful browsing should mean you don't need loads of antimalware, but I think in general it's not a good idea to on the net without protection.
Don't install more than one antivirus app. Personally I'd uninstall the Symantec offering and go for a combination of either Avast or AVG for AV (I now prefer the former); Windows Defender (which comes with Vista) and Superantispyware to run in real time (and I'm sure others will have further suggestions); and on top of those do regular scans with A Squared, Spybot and Adaware. And use Firefox.