Finding the murderer can get pretty difficult (like when all you have is an initial and the murderer's shoe size) so it can take a while depending on your difficulty level. When you finally solve a murder you can either wait until the next murder happens (in the same town) or take a job from your closest notice board. Or just dick about in the city, it's an open world after all.

I haven't done many side jobs myself yet, but I have heard that one of them (public humiliation) involves throwing a pie into a target's face so I'm pretty hyped to try one of those.

You can also try to steal stuff from other people's apartments (the devs made every apartment in the city a fully furnished area you can enter) so you can afford a better apartment for yourself and the upgrades that you can buy for it.

I had that the first time I played it. I quit after about half an hour learning all the mechanics. When I went back a second time it somehow clicked. I definitely get why people don't like it, though. You get a big city to explore but outside of the jobs you pick up yourself the world doesn't really engage you. You have to make your own fun which makes the world feel dead and empty if you're just looking around.