How do I make it so that in a multiple-level game the score will stay where it was from the previous level?
Store the score outside of the level, probably in the game object that manages which level you are on:
If there is code within a given level object that might need to manipulate the score, you can pass that information to the level when it is constructed. You could simply pass a pointer to the score integer (the simplest, most straightforward option) or you could encapsulate the score in its own interface:
Pass the a pointer to the game's instance of the single global score to each level, which they store and manipulate, and it will persist across levels as you'd like.