When I learned to play diatonic accordion, after about a year of self-teaching, I found myself taking a series of lessons from a master player near Quebec City. His teaching method was simple. We sat in chairs facing each other and he played a tune (on a one-row box in D). He then turned the low and high reeds off on his box, and I did the same.
He played the first phrase of the tune slowly. I repeated it. If I made errors, he isolated the part and repeated it, slowly, again and again, until I got it. Then we returned to the first phrase. When I played it error-free, we moved on to the next phrase, and so on. At the end of the lesson, we recorded him playing the tune, full speed, medium speed, slow speed. When I returned for the next lesson, he checked my (full-speed) playing of the tune before we moved on. We usually covered two tunes per one-hour lesson.
In the absence of a master teacher, you can use a computer with some readily available software and a recording (a CD).
For years, I used Seventh String Software's "Transcribe!" You can download a demo and try it out for a month before deciding whether this method works for you. The software allows you to slow down the playback of a tune without substantially sacrificing sound quality, and it allows you to shift the pitch of the playback. This makes it an excellent tool for someone with a C one-row wanting to learn a tune played on a D one-row. You can also "fine-tune" a recording to your box, which is helpful if you are working from historical recordings like John Kimmel or Alfred Montmarquette.
Audacity is another piece of software that you can use to emulate the master teacher's "slow-down" playing, but the software is not specifically designed for this, so I have found it less user-friendly than Transcribe. Still, it is available gratis, so it is an option for those who do not wish to invest in Transcribe.
For one-row players wishing to learning French Canadian tunes, I recommend starting with recordings that feature solo accordion, or accordion with piano accompaniment. A good place to start is Philippe Bruneau's album, Hommages.
This computer-based method lacks the element of human contact, of course. However, it has some very pedigreed lineage -- in a documentary about Philippe Bruneau, he is shown to use a reel-to-reel tape player to slow down historical recordings, repeat phrases, and learn from them!