Yes, Audacity is what you're looking for. I'm starting a radio broadcast on December 1 and we're using Audacity to record it. It's a pretty powerful piece of software at the right price.
You may also want to give Audiograbber a try. I record my sermons on a digital recorder, then transfer them to a computer (in real time, unfortunately), where they are recorded as a .wav file. Audiograbber also can covert the file into .mp3, which I then post on the web (using Podbean as a host).
We burn sermons direct to CD from the mixing board (hard copy in case the computer goes kaput in the middle).
I then transfer from CD to MP3 for our podcast and for digital archiving. I have Audacity and it is great, but often times I only need lowly old Media Player to rip the CD to MP3.