Lots of missionaries I served around had a flash drive with music on it, and speakers that would work with the flash drive or another music player. Personally, I saw this worked better than having an iPod. iPods have a greater potential to break or get stolen, but I didn't really see that happening with flash drives. You can also update them more easily, and even trade music with other missionaries. In addition, several of the newer mission vehicles have a slot for flash drives.
From what I saw, very few of the missionary apartments have CD players. However, you may want to bring CDs to use for the car.
Here are some websites where you can download free music!
https://www.lds.org/youth/music?lang=eng This website has songs from EFY, the seminary and new youth DVDs, General Conference, and more.
http://www.lds.org/general-conference/music?lang=eng This website has the music from sessions of General Conference dating back to 2006, and it has an old-timey sound.
http://setapartalbum.byu.edu/download/ Entitled "Set Apart," this album contains hymns as performed by BYU's Men's Chorus.
https://www.lds.org/music/library/music-for-youth?lang=eng This website has links to seminary music, Young Women camp songs, and selected Young Women's music.
http://www.lds.org/general-conference/conferences?lang=eng You can download conference talks dating all the way back to 1971!
If you know of any other websites, feel free to leave a comment.