Feel free to bargain when appropriate. In fact, it's expected in most markets and street stalls (though never in Cancun's ritzy shopping malls and small boutiques).
Offer half of what is asked, then work toward the middle. But don't bargain for something unless you really want it: It's considered rude not to buy once your price has been accepted.
Don't buy souvenirs made from endangered species such as turtles, crocodiles, coral and jaguars.