What software are you using to record? Chances are, you have to activate the line input for recording use. If you go to the Control Panel --> Sounds --> Audio tab --> then click Volume, it will bring up all the faders for your I/O. If the speaker icon is in your system tray, you can double click too. Once the volume window is open, go to Options --> Properties. Click the Recording button and OK to bring up your recording input volume faders. You have to check the box under Line In for that input to be active in your recording software.

As far as interfaces go, there has never been a better time to buy one. The easiest and highest quality choice would be an external Firewire interface. many of these come with 2 combination mic preamps/guitar inputs. You connect the interface to your computer through Firewire (you'll need an add-on card if you dont have any Firewire ports, also called IEEE 1394). Plug your guitar straight into it or run an output from your GT-6 into it. The device should come with drivers that will let it work with your recording software. The interface should also have a set of outputs on it that you can connect speakers to for monitoring.

Go here to see what's available:
http://www.musiciansfriend.com/srs7/g=rec/s=computer/content/cb/index

I highly recommend the Digidesign MBox, because you will then get to use the Pro Tools software, which in my opinion, is the best recording software available for audio work. If you're into MIDI, I'd check out something else.

I would not recommend a sound card device as they can sometimes be a hassle to install and use, and aren't portable. Try to find a Firewire interface within your budget.