Hi vaeel,
The vast majority of available methods for C system CBA have been written by French players, and there are quite a few of them.
The biggest issue with CBA concerns the fingering, and if you've been watching YT, you'll probably already have noticed that hardly any two players use exactly the same fingers on the same notes.
Very few people will have had the time or patience to work through all of the available method books to make a decision as to which is the best one.
I have seen Manu Maugain's method, which runs into three volumes and has a CD to accompany each one. It is probably one of the most expensive CBA methods available, although it is very well presented.
One of the main criticisms of all of the French CBA method books concerns the total omission of exercises using the 5th or inner row, and from memory Manu Maugain's book is the same as the others in that respect. In fact I'm almost certain he plays 4 row accordions. This is a throwback to the time when CBAs only had three rows of treble buttons, and to this day most CBA accordions in France only have 4 rows. The 5th row has always been available as an option, but its use has been at the discretion of the player. In short if you want to use the 5th row you have to work out the options yourself.
What this means is that if you learn the French way you'll be keeping to the outer three rows, and only using your 4th row occasionally for chords, and to avoid situations where your hand would be in an awkward position if you stayed on the first row.
The advantage of French methods is they require you to build strength in all of your right hand fingers, and the thumb and little fingers are both used as auxiliaries, with most of the playing been done with the middle three fingers of the right hand. The little finger is actually used more often than the thumb, and you'll see a lot of French players resting their thumbs on the side of the treble keyboard during normal playing. This is not "wrong", it's just the way they do it, but it doesn't look natural to non-French types who want that thumb to be one of the main players, like they do with PA.
Hope that hasn't put you off Manu Maugain's method. He teaches you to use the thumb to play scales and chords, but it is a French method after all. I will say that his fingering makes more use of the thumb than most of the other French methods I have seen.
I haven't really seen any non-French CBA method books, and I believe that Galliano's effort may be geared more towards what people expect of an "international" method. His method is just one volume with a CD but it is expensive for what it is, and has its critics.
I'm not a teacher or a pro player, but have had an interest in French accordions for a long time. Hopefully somebody with teaching experience over all 5 rows of CBA will pick up on this and be able to offer you further advice from a practical point of view.
There is no "correct" fingering for CBA. You have to use the method books to get a grounding, then adapt your fingering to the playing styles you will develop as you progress. Please don't do what most of we self taught people do. That is to get fed up part way through a method book, and decide to change to another method. That will set you back months if not years.
As George has commented, don't forget about the bellows. I did and still suffer the consequences. Most method books will show you which direction to move the bellows in each exercise. Pay attention to that, and use the air button when necessary. No point in trying to repeat a passage with your bellows already at near full stretch on your left arm.
