That should not affect your speed very much at all. Especially if you do not use one of the overweight, ponderous, suites like Norton or McAfee. Firefox and Thunderbird (and there are other choices as well) are far less susceptible to problems than Internet Explorer and Outlook. There are free AV programs like AVG and Avast! that work very well without slowing you down, and likewise spyware protection like Spybot, AdAware, and ewido.
The biggest thing that affects your online speed is the speed of your connection. The next biggest thing is that you're not running 500 other programs at the same time. Turn off unneeded Windows services (google "black viper" for a guide to Windows services). And that your computer meet certain minimum specs.
As an example, I've got a 1.5Mb/s DSL line. One of the computers I regularly web browse with is a 450MHz Pentium II with 512M RAM and Windows 2000. It's also operating as an ftp server and running a P2P file sharing application uploading at about 600Kb/s. Browsing is perfectly fine. Any machine you buy will probably be far more powerful (so long as you have at least as much RAM), and probably won't be doing as many bandwidth-consuming things at the same time.
Now (to drag this into telephony) if you use Vonage and are talking on the phone using Vonage at the same time, that's probably gonna hurt performance seriously.
Dave