A digital recorder can give good quality sound if it's set to a high enough sampling rate (assuming that you have the option to change the sampling rate). Don't expect great results from 64kbs but 256kbs would give good quality (and very large file sizes!).
It's not clear from your question whether you'll be making 'official' or 'bootleg' recordings. If you're making official recordings then the most important thing is to use a good quality external microphone (which will probably cost you several times what you paid for the minidisc or digital recorder). If you're making 'bootleg' recordings at concerts then you won't want to have an external microphone in view and, if you're stuck with using the buit-in mike, the sound quality isn't going to be be very good irrespective of the type of recorder or sampling rates used.
Chris