Harvey Birdman: Attorney At Law consists of five trials for you to defend or prosecute (or both defend and prosecute in one case). Each case is divided into 2 main parts, witness interviews/evidence gathering and trial.
So obviously, the first part of any case is going to be the crime. If there's no crime, there's no case. At the start of each case, you'll get to watch a scene that gives you some clue as to who or what is going on. After the scene, which you have the option of replaying if you would like, you get to start making your case.
You'll usually start your search in a location related to the crime. If there's no one there, you'll examine the room. Make sure to examine every location in the room at least twice, because sometimes you turn up bonus crests, which are really important to working your way through the trial. Sometimes the thing you examine is evidence. If it is, you will automatically pick it up and add it to your inventory. You'll also be able to see more details about it and possibly hear audio related to it. Not all of this evidence is actually useful in court, though. Some of it is completely extraneous.
Anyway, after you've examined certain things, other locations will open up for you to visit. Some of these have people you can talk to. If they are related to the case, you'll get a profile for them added to your evidence area. These profiles can be presented as evidence during the trial. After you've examined everything you can and questioned everyone important, the game will automatically take you to the trial portion. It will not let you go there before you have enough evidence to win your case.
Now that you're at the trial, you'll need to use all the evidence that you gathered. You'll listen to witness testimony, and then cross examine them. Their testimony is going to be divided into statements. Make sure to press each statement multiple times, even if the judge threatens you to move on. You never know when they'll change their story. You'll also need to present the evidence that you think contradicts the statements. Sometimes you'll be prompted to present evidence to prove or refute specific statements, but for the most part, you'll have to guess which evidence goes with which statement. Once you've won the case, the game will unlock the next case for you to play. You can go back and replay any of the previous ones if you want, but any bonus crests you collect won't carry over to the next case, so there's no reason to unless you just want to see them again.