What does "to pull out the throttle" mean?

Yes, it seems to.

Yes it means speed up as in bikes doing full throttle means pulling up all the gas for speed.

Well, I'm trying to figure out the meaning of "to head something out": http://idioms.thefreedictionary.com/head+out

So the whole sentence means: Keep the boat moving and speed up?

Head the boat out here means steer the boat away from the dock towards the open sea. The boat's heading is the direction it is going.

And "I headed out the car and we were on our way." means s/he moved the car away from where it had been previously standing, right?

Yes, but headed implies moving towards a destination. You wouldn't use headed to refer to moving the car nearer to the door in order to load things in it.

AnonymousI headed out the car and we were on our way.

To me this sounds like

I started driving the car toward our destination, and were on our way.

Personally, I don't use "headed out the car". It sounds a little strange to my ear.

CJ