Detective: Detective stories can be private detectives/investigators or law enforcement detectives. These range from light, near cozy style to dark mystery, noir and beyond--pretty much a catch-all category. These are not police procedurals --that is a separate sub-genre. These stories focus on the detective. Honestly, even cozies fit into this category since many of them are using amateur detectives. TV version of the standard detective story would be: Magnum P.I. for example.
But where to start? I'd say go for Michael Connelly and his Harry Bosch series, beginning with The Black Echo. I'd characterize the series as a standard detective series that is squarely in the middle of the light to dark spectrum (cozy being light, noir being dark).
Police Procedural: The main character works for the police, or is an attorney, or is a federal law enforcement officer, etc. These stories focus on the procedures in solving a crime.
Hill Street Blues is a great TV example of the procedural--in fact, the series was based off Ed McBain's 87th Precinct novels which I'm going to recommend here. There were also plenty of TV movies based on these novels. But if you enjoy seeing how the justice system works or how police go about solving crimes, then procedurals are for you.
Where to start? Ed McBain--he wrote a ton of books based around the fictional 87th Precinct. They are great reads and pretty fast. You can start almost anywhere with them despite them having recurring characters. McBain wrote these from the mid 1950's all the way until around 2005--so, there's quite a bit of material there.
Hardboiled: I think these are my favorites. Character driven and a lot of attitude. You feel the world, see everything from the character's point of view. The main character is cynical, but has a moral code. The world the character inhabits is visceral, gritty, and typically urban.