Fallout

I have found the Fallout series to be good at integrating good music into an enjoyable game.
First you should know that this game was released back in 1997. This is a pretty good game if you are not worried about graphics.
Fallout is a post-apocalyptic computer role-playing game that was developed and published by Interplay in 1997 and is the first game in the Fallout series. The game takes place in the year 2161 on the West Coast of what used to be the United States of America, particularly in California.

You take on the role of a "vault dweller" from vault 13 and it is your job to either save the inhabitants and quite possibly humanity or sit back and do nothing. There are many choices that can be made in this game and all of them will be reflected in the games ending.

And apparently you can pick up the trilogy(fallout 1+2 and tactics) for under $10 on amazon. It's worth checking out.