Does movement end when you land on an open gate?
There are two different forms of movement in Arkham Horror.  These are described on page 6 of the rulebook.

They are:


Arkham Movement
Other World Movement


Each investigator must choose which they will use based on where they are located at the beginning of their movement.

If you are using Arkham Movement, use your speed points to move as you will.  Of course you need to stop if you fight a monster.  Other than that, there are few limits.

You do not use speed points to go through a gate.  Going through a gate is typically done during Phase III: Arkham Encounters