“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 trying for 30 minutes. Now you're throwing it away. You think: 

Ah well... I gave it my best shot.

Ah well

This phrase is a variation of "Oh well":

Oh well, it'll come back to me in a few minutes.

"Ah well" sounds a little more disappointed, while "Oh well" sounds more optimistic. "Oh well" is more common. 

give (something) (one's) best shot

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.