one question dude - does your laptop have an asio soundcard?
if not, it makes the process a bit harder, because without one direct line recording isn't possible on a laptop.
my laptop hasn't got one, but i've found ways round it with fairly good results.
first of all i guess it also depends on what you mean by recording your own songs...do you play an instrument, do you want backing tracks that you can fully create yourself, or did you just want to record you singing over someone else's work?
i use a program called fruity loops...it is primarily meant for creating dance music, but i play acoustic guitar and i write backing tracks on this program, record my guitar (its electro-acoustic so i just plug it in to laptop) with some audio recording software, and then paste it into the song and position it in the right place....its fairly easy and fruity loops gives you loads of effects tweak the sound to how you want from a simple eq to chorus emulation, reverb, and delay banks etc etc
you can get fruity loops for free if you know where to go, and in terms of microphone you want a good dynamic mic, i use a SHURE SM58 cost arounf £60 but defo worth it
hope this helps mate