Hi guys, sorry for the super delayed response, I appreciate all of the replies!
I wondered about the P Trap, but I thought maybe it was under the ground? but that wouldn't make sense...how would you clean it out? Hmm..
To answer someone's question about the green hose, yes it is the water softener drain, but it goes through a long series of pipes before it gets there so I was hoping it wasn't too bad.
What I'm doing right now is just pinching off the furnace drain (the clear one at the bottom) and turning the furnace off whenever I do laundry so it doesn't back-wash through.
I'll be calling in a plumber I think on this one, thanks for the advice