The Back Door and Jai Passe Devant ta Porte should keep you busy for a month or two. This gives you a two step and a waltz, both very basic, to work on. Play one of them until it drives you crazy, then switch to the other for relief. Then repeat, repeat, repeat.
When you can play both of them clean at a comfortable tempo, pick something a little more complicated. Thus begins the process that should never end.
Here's my idea on the subject: The best songs for a beginner to play are songs they love so much that they will do anything to be able to play them. Learning to play is a long, slow road. You will need patience and perseverance. Having songs you are passionate about is a great motivator, and you will need that motivation to sustain you.
No matter what the song is, there will be a simplified way you can start out with it. Don't try to copy every little embellishment you hear in the recordings of the great players.
But, it's true that some songs might be easier to learn than others. A good waltz for beginners that comes to mind is Love Bridge Waltz. I don't remember if it was the first waltz I learned, but it was definitely the first waltz I learned to play well enough to play at gigs early on.
I would recommend looking up the stuff by Allie Young. There is a goodly collection on Youtube and it's mostly accordion only. His playing is really clean and you should be able to listen with it slowed down (Amazing Slow Downer) and figure what he is doing.
Vidrine, you're in luck! Although we don't have many old masters left alive to play it right in front of us person to person, we do have hours and hours of YouTube Cajun music video that's available today and it's growing by the week! Hell I recently found the complete collections of Amede Ardoin and Iry Lejeune on YouTube. That selection of Allie Young that John A posted is excellent. I went back and played along with every song on there and they are done "RIGHT". Like Octa Clark said, "You can't go wrong if you play it right." I would be impressed to hear or see any beginner accordion player break out with one of these older traditional songs and play it accurately and at good tempo and play it from the heart. Stick with the classics and don't get caught up with too much of these modern day professional players styles. Many of them actually missed the mark in my opinion. I always say, life is long. You have plenty of time to learn things and learn them right. You'll have plenty of time if you take care to live a long life that is. Don't be in such a rush to present yourself and what you've learned on accordion until you have something worth presenting. Don't worry, there are plenty of fools and dancing chickens that will fill in the gaps and put on a show with worthless talent while you lay low and learn the good ole music the way it was meant to be played.
Oh, Vidrine. Would you happen to have a "D" accordion? If you do, seek out Amede Ardoin's recordings on youtube that are under the record title "Mama I'll Be Long Gone" the complete collection of Amede Ardoin. Every song is done on a "D" accordion. He plays all 3 positions, but still on a "D" accordion.
Here's the link below and I'll attach it at the very bottom in links also. The album list shows up on the right. There are 34 songs. 2 or 3 of them are repeats named incorrectly, but a treasure trove none the less if you're willing to bear down and give Amede a try. If I were to have a son who wanted to learn accordion under my guidance, after he learned to play the triangle, he would learn on a "D" accordion first and learn Amede's songs first. This would open him up to all of the con artist that have copied his songs since time memorial. I'd have him start with the root of the tree instead of screwing around fighting with all the got dam branches so to speak.
https://www.youtube.com/watch?v=LaiR4uin9qE&list=PLFoD3l_5JLlAs1yz-luqKpmNTjaNKycPi