As a computer programmer, one of the first rules you learn is "Garbage In, Garbage Out". Once the correctness of the data - or in this case the quality of the sound - is gone, there's very little that can be done to bring it back into a usable state. Keep this in mind when deciding how best to spend your money which can easily disappear on a whole myriad of different items from instruments, mics, cables, pedals, outboards, rigs to DAW software.
For the moment I would prioritise your setup like this:
- Effects (dry or wet)
- Noise removal
Firstly, get an acoustic pickup that has low (acceptable) noise. You can test this out by listening through a good set of headphones before putting it through a computer. If the signal you're getting at the very source isn't good enough don't try to fix it once you get it into a digital recorded state.
Second, if you're going to be recording music on a PC it will be worth considering getting a soundcard for that purpose. If you're using a laptop you could get an external soundcard e.g. DMX 6Fire. There's nothing worse than not being able to get a good recording because your soundcard isn't low latency or powerful enough to keep up.
The only time I ever put distortion effects on an acoustic guitar sound it didn't come out too well but in principle there shouldn't be a problem. You could either buy a purpose built guitar FX pedal e.g. Digitech Grunge, Boss Metal Zone or use software on the clean acoustic signal later e.g. the Fruity Blood Overdrive plugin for FL Studio. I'd say the pedal route should be if the 14 yr-old will some day gig or otherwise want to tailor their rock distortion or whatever and let it be portable.
A good DAW system will come with noise removal software but (a) you're looking at spending a lot of money for just recording an acoustic guitar and (b) it can still have unwanted side-effects compared to the original. I found when I needed to use noise reduction because of a badly recorded source signal the result had a very muddy sound. That's why I'd recommend considering the noise removal element last and instead try to get a system working that doesn't need it. Regardless though, if you choose to use software effects to change the sound away from an acoustic guitar you'll need a DAW that can handle a plugin to do so. I mentioned FL Studio but there are others (including free ones) mentioned in the wikipedia link above.