First of all, I'm by no means an industry professional, so I don't know much as far as the technical side of things. I'll try to help though.
Unfortunately, as a musician I can tell you that almost all the software commercialy availabe is going to use technical references, aka the industry standard lingo.
However, most of it is actualy surprisingly easy to get into. I recommend a simple program like the more basic Sound Forge programs, or my personal preference, Cakewalk Home Studio. It's more than good enough to start with, and will do everything you need.
You can have a more hands-on software environment with those programs. In other words you can actualy very carefuly make your own adjustments to volume level, equalization of individual tracks, you can make automated envelopes for volume, panning, effects processes, etc. It seems very overwhelming at first, but they come with very handy tutorials that are more or less in laymen's terms.
Like anything you love, you have to work at it, and stick with it to learn. It takes quite a bit of dedication and work to learn the ins and outs, especialy for mastering (which is essentialy what you're talking about doing).
If all else fails, you can also simply re-record the performances with the new software as well. Digital home recording is one of the most cost effective means of gaining control over your music. That might not be feasible though if these are old recordings that can't be reproduced in any fashion for whatever reason.
Note that if you are going to use Cakewalk software, I recommend using a WDM driver compatable sound card. M-Audio makes some nice ones, and surprisingly affordable ones. The Audiophile 2496 would be a good card to start with if you don't need alot of I/Os. It's capable of 24-bit 96kHz recording, which is what you want for more or less professional quality sound. The audio software will let you compress back to 16 bit for burning to CD as wav files without any loss of quality.
P.S.
Have you considered playing the mini discs back from a mini disc compatable multitrack rig routed to the PC, and trying to EQ out some of the buzzing pre-mastering, and then work on it more within the PC software environment? If it's that noticeable, it probably needs alot of fine tuning anyway, if it's salvageable at all.
I hope this helps somewhat, atleast, and I hope I've correctly understood your problem.
Good luck to you!