Hi there
Go to the music app and start a track. Touch the bar near the bottom that has the currently playing track's details and it will grow up the screen to provide you with more information. At the very bottom of the screen now will be three icons. The middle one is two arrows, one above the other, each pointing in a different direction. If the icon has a grey box round it with a really small '1' in the top right corner, tap the icon and the number and grey box will disappear. This option is the one that repeats the album, repeats a single track or, when not with a grey box round it, just playing tracks one after the other normally.
With regards to songs that are downloaded from your computer, you may have to download them again.
However, to check, whilst in the music app, click the little musical note icon at the bottom left that is marked 'My Music'. If you have any downloaded music on your device, it will be listed here. For reference, the new update to Apple Music didn't remove the music I have on my iPhone.