I took some code from my libgdx project to get flexible screen transitions. Here is the Github link.
There is another a simple approach, if you don’t need special effects, where both screens are visible at the same time. Take a look at this tutorial.
The approach i follow is basically from this cool Book:
Download the jar file and add it as a jar dependecy or just use the source code.
How it works:
The FadingGame class extends the standard libgdx Game class, so if you use the Game class already you just need to replace the existing one.
Find more information on github.