Yes, you need a driver to take you to the store to get a microphone.
Here's the deal. I'm not sure where you are in your seriousness level / budget. However, recording direct guitars sucks and use the stock soundcard sucks even more because of latency. While you could probably get your current rig to work (in a way that resembles that episode where McGuiver built an airplane out of duct tape, bamboo, and a concrete mixer), it's not ideal for anything remotely resembling "serious" recording.
Now if you just want to goof around, I can help you with that too.
If there is any evidence for the existence of Satan, this is it. The 1/4 to 1/8 adapter is the root of all evil.
In all seriousness, there is nothing "wrong" with these adapters, they have just have a tendency to fail. It's pretty much inevitable that all of these adapters will fail not long after purchasing them.
Assuming we are just trying to get your current rig going, you need to start with the amp. Every amp is different. If it has a line out, you could use that. If it has a headphone out, you could use that. It's possible that the volume of the amp will control the volume of the line out, but it may not. It just depends on the amp.
I'd start with running the amp into the "line in" on the sound card because the signal coming out of the amp should already be line level. You'll need to active this input on the sound card and make sure that it's volume is up.
In Audacity or whatever, you'll need to select this as the input.
It should be relatively straight forward, but much of it depends on the sound card itself and the amp.
Brandon