“Ah well... I gave it my best shot.”
Your old music player broke. You tried to fix it, but you weren't able to after for 30 minutes. Now you're throwing it away. You think this to yourself.
Ah well... I gave it my best shot.
This phrase is a variation of "Oh well":
"Ah well" sounds a little more disappointed, while "Oh well" sounds more optimistic. "Oh well" is more common.
When you "give something your best shot", it means that you try as hard as you can to do it, even though you're probably going to fail:
A: I can't do it. I'm not good at this kind of thing.
B: That's OK. Just give it your best shot.
People usually use this expression to talk about someone trying, but failing, to do something hard.