I am using JUnit for the first time in a project and I'm fascinated by the way it is forcing me to restructure my code. One thing I've noticed is that the number of objects I've created in order to be able to test chunks of code is significantly increasing. Is this typical?
Yes, this is normal.
In general the smaller/more focused your classes and methods are, the easier to understand and test them. This might produce more files and actual lines of code, but it is because you are adding more abstractions that makes your code have a better/cleaner design.
You may want to read about the Single Responsibility Principle. Uncle Bob also has some re-factoring examples in his book called Clean Code where he touches on exactly these points.
One more thing when you are unit testing. Dependency Injection is one of the single most important thing that will save you a lot of headaches when it comes to structuring your code. (And just for clarification, DI will not necessary cause you to have more classes, but it will help decouple your classes more from each other.)