The quickest and easiest way to start recording your guitar, especially if you are on a very tight budget, is to buy one of these entry-level dedicated guitar-USB interfaces. They have been designed to quickly and easily bridge the gap between your guitar and your computer. So you plug your guitar in one end and connect the other to a USB port. Some even come with guitar recording software. Even if they don’t, you will easily be able to record and edit your guitar in Audacity or any other DAW recording software. You will be recording the ‘dry’ signal of your guitar but will be able to use amp simulators and other effects on your computer to create a big fat guitar sound. And you are not limited just to the amp you have.
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If you are also interested in recording your vocals or other instruments, then you might do better buying a full-blown audio interface so you can record your guitar and other instruments and microphones as well. If you want to know more then read about what is an audio interface, and then how to choose the best audio interface for your home recording studio setup.