Funny, I have a lot of statement necklaces acquired over the years, and lately I'm finding myself "off" them and craving instead simple, delicate, understated pieces.
But I can still give you tips.
For me the #1 requirement is scale. Because I'm short with a short neck, the scale has to be just right. Too big, it can overwhelm. Too long, it can fall all wrong. So I look for medium-sized and on me that's just-right "statement" size. Scale probably eliminates 90% of the options out there. Scale is also the reason I rarely buy online. I have to try the piece on.
Absolutely look at art fairs -- but don't limit yourself to just that. There are a lot of fun inexpensive pieces at your regular retail shops like LOFT, Jcrew, AT, and BR too. Heck, one of my favorite pieces was purchased at a kiosk at the airport. It's cheap as can be but gets so many compliments. Leave no stone unturned!
Think about your tops and what style will work best. For example, I wear a lot of V-necks so bib necklaces work nicely to fill in the space.
Think about what colors you like, and that would make a great accent with your wardrobe. I have lots of pieces with white, because white near my face lights it up. I have a lot of pieces with green, because it accents my eyes. And my favorite piece of all is blue.