As with any adventure-game locale, look everywhere that you can and talk to everyone. And if you can pick something up, do so, because you will almost certainly need it later, no matter what it is.
Learn to use the in-game camera function; not only will Gabriel and Grace move more quickly around the game's locales, but certain puzzles will be easier to solve if you look around with the camera.
Once you get the fingerprint kit, you can look at or inspect a suspect's inventory items, but don't pick them up until you dust them for fingerprints.
Save the game after the first murder occurs. The game will then shift into real time, and Gabriel can wander around, talking to or spying on the suspects, but only for five minutes. If you go back to the save point, you'll be able to take Gabriel on a different route and learn different things about the case.
On the game map, the Hint button tells you where you need to go to finish a section of the game. But try to go to every location, because you never know whom you might bump into or what you might learn about the case.
Developer Cameo
Locate the bookstore window that has the book on the Holy Grail. Look in the window next to the door to see Jane Jensen, the woman that developed the Gabriel Knight series of games. Additionally, examine the book that Gabriel authored about the Voodoo Murders. Gabriel will say "Wow, isn't he some famous author or something?"