A version of this question is asked every day somewhere on the internet it seems.
Today on Yahoo! Answers, somebody asked a pretty common question, which was:
Q: I'd like to set up a home recording studio so I can record guitar parts and maybe vocal parts, and I want them to have really good sound quality. What would I need to do this?
A: Hi Chris, Definitely go download Reaper (by Cockos). It's free for the 60-day trial period (yup - I said "60 days") and then only $60 if you want to buy a license. That is until you start making 10-20-grand a year with it, at which point they prefer you buy the $220 "commercial license."
But it's all the same exact software. These guys are awesome. Oh, and their forum is super active. Questions are usually answered within 24 hours by several people.
Anyway, that's your software. As for the mic, don't start out with a big expensive mic. I say try out a USB mic (like the Samson Q2U) for about $59. That way you won't need a separate interface box to plug a normal mic into. If you're not happy with that, then I recommend a small-diaphragm condenser mic (I use a Shure SM-81) along with an audio interface like the Focusrite Scarlet 2i2. I've been using one of those for the last few years. I love it.
It might help to take a look at our 5-part post series called "How to Build A Home Recording Studio".
Hope that helps!
[update 2020] Since writing that, I have some more resources that I would have included for this answer. Those include two home recording started kits/bundles that B&H Photo-Video-Audio put together according to our specs. You can read the details about those two bundles here:
I also created a mini-course called "How to Create a Home Recording Studio." This will tell you all about the gear you might need for 4 different budget levels.