Hi Geeman, welcome!
Yes, the TASCAM 02 can do this, although it is a bare bones machine. Basically, look at it this way: with a 4 track you can record 4 seperate audio tracks at different times. I will just give you a "for example" On trk 1, you can lay down a drum track. Then you can go back and record a guitar track on track 2 while listening to the drums on track 1. When that is done, you can go to track 3 and record another instrument while you listen and play along with tracks 1 & 2. Then you can grab a mic and sing the vocal track on track 4 while listening to the music on tracks 1, 2 & 3 simultainiously.
Now, if you need more tracks, don't despair! There are little techniques that you can use like "ping ponging" that will degrade the sound a little, but will give you more tracks for recording. But it's a little too soon to go into that. IF you are as new to this as you said, you may need to chew on the info above for a little bit first. But to answer your question, yes, a 4 track will work.
Also, you can use your computer as a multi track recorder, but I will let someone else tell you about that because I have never used my computer that way. Other guys around here will probably chime in on this one...
Good luck, and I hope this helped!
Brad