The software won't really change the recording quality, if the mic is bad, there's almost nothing to do about it.
There are some software with noise reduction, but those filters don't really work that well.
What I would suggest you to do (if you wanna be a little bit serious about this), is that you purchase a small 5-track mixer with an XLR-input, the Behringers cost about 50$. With a mixer you can connect MUCH better microphones. Then you can connect the mixer to your computer with Aux to minijack.
So, what I would reccomend is getting:
And some dynamic microphone.
There also is an USB-microphone that could work, here: Samson Q10.
Should also work allright, but if you purchase a mixer you will have a wider range of higher quality microphones.
But if you just want to have some fun, I'd just reccomend that you go with the USB-mic, it will give you highly acceptable quality.