I need to press one button, each time I press it, I need a new/different output, 4 total outputs (Minecraft)

Solution 1:

Probably not the most compact solution, but one that should work in every release since some beta version and probably also the future:

Since these images aren't enough to rebuild it, here's a structure file.*

The size is 6x6x6 plus an input line that sends a two redstone tick long signal.

How it works:
The stone block and the three glass blocks get moved around by the pistons. Under each of them there's a redstone torch, but only the stone block conducts its signal, which can be taken out with a repeater or redstone dust. This principle is called a "piston tape", because you can basically write binary data into it, this is just a really small example with the data 0001.
The pistons are activated with proper timing using a small torch tower, because the wiring got extremely messy when I tried to put input and output on roughly the same height.

*That file is on my Google Drive, but it shouldn't get deleted. If I accidentally delete it, contact me (contact details on my profile page) saying that the file is in my November 2018 backup and link this answer.