Thank you for your answers.
I looked into shift registers and it seems that's what I need. They also seem pretty simple to use.
I couldn't find any 2N2222 transistors, only 2N2222A ones. Would these work as well?
So, if I got it all right, it's going to look like this:
Arduino connected to 4 shift registers that are connected to 20+ transistors (base). Between the registers and the transistors are resistors.
Transistors are connected to LED strings (collector to negative).
12V power source goes to LED string (positive), shift registers and arduino.
Ground goes to transistors (emitter), shift registers and arduino.
Few more newbie questions (sorry I'm a software engineer with a limited knowledge in electronics and electricity)
Why do I need transistors?
Does it matter how many LEDs I have in each string?
What are my 12V power source options?