With respect to Time Machine, you might want to see if anything here helps: Time Machine troubleshooting on Mac - Apple Support .
If you can't get Time Machine working or you want to make extra backups via other means, you could use Carbon Copy Cloner or SuperDuper! to selectively back up the folders containing photos and music to an external drive. (I have Carbon Copy Cloner, and am assuming that SuperDuper! also has this feature.)
Another possibility would be to copy the folders containing those things to an external drive using the Finder. The question there is whether the Finder might choke on the number of items you are copying and throw up an "Error -36". In the old days, that used to be a possibility with very large Finder-based copies, which is why some people got into the habit of using other tools seen as being more reliable.