To accomplish this week's task, all you need is some software that'll "rip" from CD and allow you to edit. "Rip" the tracks off the source CDs, make your edits, then burn a new CD of your edited stuff (watch for copyright violations!!!). I would use Sound Forge, but that's kind of high end if all you want to do is a small edit for a dance track. Cakewalk's Pyro will do this as well- I think it's only about $50. There are numerous other programs that will do this as well. The 2 I mentioned are the ones I'm familiar with and have used.
To accomplish this month's task, you will have to find a "line in" to get audio into your sound card. I did a quick search on your HP machine and it appears that it doesn't have one. All is not lost, though, because you computer has a USB port. There are numerous USB audio interfaces on the market right now. Something like a TASCAM model US-122
or an MAudio Audiophile USB
unit will meet your needs. Both of these units will allow you to record audio to your hard drive via the USB port on your computer. To record the tapes, just hook the output of your tape deck to the inputs of the USB interface you buy, call up your recording software (again, I would use Sound Forge, Pyro, or Cakewalk Pro Audio which is my primary studio software), hit "record", then hit "play " on the tape deck. Recording your kids music lessons would involve a microphone and stand (2 if you want stereo!) into the USB interface, calling up your reocrding program, and recording it.
To accomplish this year's task is quite a bit more involved. You would have to be more specific about what kind of band your little prodigy( ;) ) is involved in, what the instrumentation of the band is, etc. You can spend tens of thousands setting up to do this type of work. If you want to make any kind of CD of the group that is anything more than a REALLY rough demo for posterity, you'll have to spend at LEAST 2-3K to come up with a minimal setup that will allow you to record a full band. Now, if there's no drums involved and your kid's band is a simple two or three piece, then you could do it for maybe 1K (or even a little less) assuming every one already has their own instruments, but it'd have to be pretty simple (or else not tracked live).
Hope this helps.... :D