Deciding to purchase your own gear can be the edge of a very deep abyss, be careful haha. Since you are starting, it's my recommendation to start slow. Focusrite makes the 2i2 studio bundle, which would be a great start for about $200-250USD. The 2i2 is an interface with two good sounding preamps. The bundle also comes with a decent large-diaphragm condenser microphone, an xlr cable, and studio (flatish-eq) headphones.
At the very least, you should look into a proper interface like the 2i2. Using an interface will always be inherently better than just using the soundcard, because of the soundcard's proximity to the other components of your computer, and the signal noise caused by that.
You'll want an actual DAW, but something like Audacity, Reaper, or MU Labs will be great, and they're free. If you want to pay for more, Fruity Loops is great for making beats, and Ableton is my personal favorite. Ableton is expensive if you go all out though.
OK! As for mics, I agree with Schizomorph, you should consider how you will hold the mic. If you see "condenser" or "ribbon" when reading about a mic, definitely get a stand and even a shock-mount. These mics will be very clear and accurate sounding, which isn't always great for rap. Condensor mics (NOT Ribbon) will need phantom power going back to the mic.
If you want to hold it, get a dynamic microphone. The dynamic capsule is slower than, say, a condenser capsule, so it will 'round' off transients. This means a bit less than high-end, and a little less noise-transfer from handling the mic. I find dynamic mics much more forgiving in terms of recording environment.
Specifically, I recommend (from cheapest to most expensive):
AKG D5 is pretty cheap and good for if you want to use it in a live environment as well.
Shure SM58. Similar to the D5. Very dependable.
Audio Technica AT2020. This one is a condensor mic. It is quality sounding and would be good for an in-studio environment.
AKG 214. This is another condenser, but it's more thick and warm than I other condensors I've worked with. This one also handles loud voices very well.
There are a lot of mics out there, but I hope that's a good start.