Advanced keyless card. Is it supposed to lock my doors?
 

What I would like to know is if my car is supposed to lock its doors when I go away from it or do I have to push the rubber button to lock it every time.

I believe that you have to push the rubber button on the outside handle.

Push the button...
and you can't lock it like that if your engine is running (ie if you forgot something and have to run back inside the house). You have to pull out the real key and do it manually.

I do believe, however, that there is an a way to turn on having the doors lock automatically when you (with the key) move away from the car. I think I remember the sales guy mentioning this when I picked up my 8. I will check the manual when I get a chance.

maybe the trunk is open?

According to several other people with this option, you're correct. You have to get it programmed by the dealer to auto-lock when you walk away. The fact that this isn't the standard configuration is bizarre in my mind.