I kinda wanna start with this one, since it's the most recent entry and the plot sounds the most interesting, but I have a feeling that it will spoil me with more cutscenes than the others (or so I hear) and possibly better puzzles, and if I like it and decide to play the others, I may have difficulty going back to the previous installments...
So back to the title, do I need to play this series in order? Or...?
While each game has it's own main story, they're part of a trilogy with recurring characters, a sub-plot running through all three games (which comes to a conclusion in this game) and previous game spoilers (such as an early scene in the second game spoiling a significant plot twist in the first game)
Although the games can be played out of order if you really wanted to, it's naturally better to play them in the correct order
Which game is best is simply personal opinion (as seen in other threads, not all players agree about this and have quite divergent views) and someone else's opinion may be different to your own
Professor Layton and the Unwound Future Illustrated Walkthrough http://professorlayton3walkthrough.blogspot.com/
Also, this game has some flavor-text references to the first two. If you're like me and nearly murder the screen of the DS looking for those hint tokens... yeah. They're nothing major, only references to locations, but sometimes even a truly minor spoiler is still a spoiler.
My entire existence was constructed by a sociopath in a sweater vest, what do you suggest I do? - Dr. Saunders, Dollhouse Season 2 Episode 1 "Vows"