When you open a door for the first time and reveal a room, the app will usually say that you may move the investigator 1 space into the room. Now lets say you performed a move to get to the door, and it was 1 space, meaning you still have 1 space of movement left. If you open the door, and move the investigator into the room (which is your second action, being opening the door), does this mean you still have the movement from earlier left over, and can go one further space into the room?

Exactly. Movement can be interrupted by other actions, so, you are definitely allowed to move 1 space, explore, and then move another space

Awesome - I just wanted to make sure the movement portion of the explore didn't interfere with the movement I had left from the move action.

The game doesn't know that you have already make one movement, or two, or none, so I'm guessing that means you have 1 free movement step to go inside. If you previously had spare movement steps to make, I think you can move even further inside. Am I correct?

Yes, you can move as instructed as part of the explore action, then continue with movement.