How to progress? Genuine question

Hi there, apologies if this is not the correct place to ask.

But can you help me understand how to adequately progress? I have 25h of gameplay, but don't quite understand how to make everything evolve.

Would it be related entirely to dumping cats to the correct places only? Because I've finished a couple times the runs to the Cemetery and the Junkyard, but it never seem to progress anything. My cats continue to be lame.

If you could help me without spoiling anything, I'm at kinda of a blind spot now.

Thanks in advance!