Carol:
Rich up front I'm not familiar with mini-disc recording.
I've digitized quite a few LPs, cassettes and open reel tapes that go back 40 years or so.

There are some previous threads also labeled "Tech" which may prove helpful. I know there are some minidisc fans out there who may urge you to somehow employ this.

To my mnd the simplest way to go about it is to feed an analog signal directly into your computer, utilizing software which knows how to 'save' the digitized music onto your hard drive:

You need for the computer:
1)a 'line-in' receptacle, which either comes on a soundcard or on the more recent motherboards. The line in receptacle expects to see stereo or mono analog sound and passes the signal to a converter so the computer will see this as digital.
2)A program which will process the digitized sound. I use "cooledit" which is no longer available, but is a wonderful program. There are programs which are free downloads which can do this. In general the sound is 'saved' as a wave file (.wav) The program will ask you how you want to sample the music and generally give you a choice involving sample rates and stereo/mono. If you have a CD burner in your computer, the software that came with it might have a sound editor in it. Typically the Nero software provides a nice one.

Of course, what you're sampling dictates how much computer power and memory you require to preserve it. For a start make sure your cassette player is plugged into an outlet or has fresh batteries. You want the sound going in to be as good as possible.

You will need some kind of patch cord. For a hesdphone to line in you requrie a mini-jack at either end. They are inexpensive and available at many big box stores in the audio department. If you intend to use stereo, make sure it's a stereo mini-jack at either end of the patch cord.

At the high end, if you are sampling stereo music, you will select 'stereo' and a sample rate of 44100. If you are sampling monophonic spoken word, select 'mono' and a sample rate of 22050.

High quality music takes up about 10 megabytes of storage per minute of music. If your original tapes are not super high quality, you won't lose anything by sampling at a lower rate, thus you can get more music into memory.

The big advantage to this is that you can play around with the computer and use your ears to judge whether you're getting what you want.

Hope this helps.