I started as soon as I could and went and explored most of the map. I have about 65/95 unlocked. While many of the cars look very similar to each other, you can always get a good idea if you note the front grille variations and especially the different marques and emblems on the rear end. Also one car may be strikingly similar with a variant of a spare tire on the back, etc.
If you visit the army base on the far southeast of the map there are some rare cars parked out front (once you get past the gate) and if you traverse towards the north (middle of the map) where the rich folks live you can find an assortment of different cars in their gated properties.
Similarly at every crime scene I get into all the cars at the scene that the reporters and police are using BEFORE I go to the actual scene. Also if you see a lot full of cars it doesn't hurt to pull into it and start jumping into various vehicles. Another helpful tip is whenever you drive to a new location or even a street crime, try to jump into cars parked on the street before you actually start the cutscene or interrogate someone. Using these tips you can gain access to cars like the Studebaker Commander, Coroner's vehicle, Ambulances and Paddywagons.