When I open a new tab it opens with babylon.com -- how do I make that go away?

Looks like the Babylon application got installed along with some other program. You will need to uninstall it. Unfortunately, there does not seem to be a straightforward way to do this since Babylon comes in several avatars. See the discussion in this article where other users have provided ways to successfully remove Babylon: https://support.mozilla.com/questions/871318.

Go to the add-ons menu, go to extensions, at or near the top of the list, there should be a bar that says babylon on it. Click remove, restart now, and viola! No more babylon!