And we don't have any idea what to do with it. We were on vacation in Ireland, walked into a music store in Skibbereen during a close-out-going-out-of-business sale. A two row button Serenelli accordion had my wife's attention and we walked out the door with it at what we thought was a bargain. Don't even know. Anyway, how can we know what key this box is tuned to? Can anyone suggest a learning book and where to purchase one? We are both proficient at reading music.
Before you can look at tutor books or videos, you need to know what system the box is tuned in - which means establishing the key of both rows.
If the box has 23 buttons on the right-hand (treble) side, then the 4th button away from the chin-end is the tonic or "doh" - when pressing the bellows (closing them). If it's a 21-button box, the 3rd button is doh.
So open the bellows a little by playing a note or using the air release valve (on the bass side, on the side of the cabinet above the bass buttons) and then press your doh button while you squeeze the bellows closed and test it against a piano or other instrument.
Do the same on the other row.
Since you bought it in Ireland, the box is most likely to be in B/C (C being the inner row, B the outer), or C#/D. But there are many other possibilities, including D/D#, C/C#, or D/G, C/F, G/C and the list goes on.
There are a few tutors for B/C box and quite a bit of stuff for D/G. Let us know what you find out and we'll point you in some direction or other.
Serenellini make good boxes. Do you have the name quite right in your post?