To simplify this for everyone, movement through terrain is best described this way; The territory you are standing in does not count when it comes to movement. Assume you are standing right at the border. The first territory you encounter determines what type of terrain you are moving in. If there are no terrain features you have to cross in the first territory you cross then you can move into a second territory if your unit has movement points left. Cavalry is the only unit that could move through a territory with terrain (mountain) and into a second territory.
On the other hand, there are restrictions involving blitzing. For instance, you cannot blitz out of a city. That means that if you moved into a city on combat movement phase and successfully conquered that territory, then you can’t subsequently move into a second enemy territory as a blitz.
You can’t blitz into a mountain territory or over a mountain border. That means that you can’t attack an enemy territory with no terrain on combat movement and then blitz into a second enemy mountain territory and conquer it. You could, however, move through a friendly territory with no terrain and then attack a mountain territory as that is not considered a blitz. The first territory you entered did not restrict your movement.
Thanks so much, GHG! We really needed an answer for this one!