All you need for the moment is a mic and a headphone. The mic that come with the pc can't handle high vol (SPL) and your singing will sound distorted. So just buy any stage mic. Then make sure you have an 1/4" to 1/8" adopter for the mic to be used with the sound card of your pc.
Use the headphone rather than the speakers because the sound from the speaker will be recorded thru the mic. this will produce a ghostly sound in your recording.
You need a proper recording program and there are free one. I use KRISTAL but most people here start with Audacity. here is the link.
http://audacity.sourceforge.net/
Learn how to use it. And here is a good link
http://www.daniel.uklinux.net/tutorial/
hope this help. and the whole set up cost you less than $50. use the set up until its time for you to up grade to buying things like a condenser mic, pre amp, pro soundcard etc