I always have to press the right arrow button to skip to the next song in my playlist manually though.

As others have said, it sounds like you have "Repeat this track" turned on.

What others didn't say is how to fix it.

From the main menu select "Repeat", and then select the desired option.

You probably accidentally turned it on by pressing the number 7 key on the remote control. So that is another way to fix it. Cycle through the repeat options by pressing 7.