On the street, you can't just tell the good guys from the bad guys. You need to be vigilant and watch everything that you can see. You can't act immediately as you need to wait till they commit a crime. That is how it goes and you can't do anything about that.
However, if they are escaping on a vehicle then you know what to do since you are given authority. You need to take them down if they won't cooperate with you. More that likely, you need to take them down since these gangster are heavily armed and would not want to get arrested.
In this game, you don't have a major objective since there are several objectives in a mission. You need to do it over and over again until you complete a mission to be able to proceed to the next level. This would involve you to chase down enemy cars and destroying by shooting at it. Try not to collide with any car as that would damage your car too. There are no way to repair your car while you are on the road, so make sure that you won't damage your car. The game gets harder and harder as you unlock new levels.