Can I add my own words or phrases?

Yes. To do so, you first need to create a personalized vocabulary list:

1/ Click on "Create personalized list" on your vocabulary page.

2/ Give your list a title then click on "Create".

After having done this you will see the option to "Add a word" when you access the list. Using this allows you to type in your own words and phrases. Incidentally, if what you type already exists on Babbel, any matches will appear and you need only click on them to add them to your list.