The Honolulu Humane Society maintains a list of dog-friendly parks & beaches. Check out the "Animal-Related Information" tab on their website
E Komo Mai! | Hawaiian Humane Society. (If your dog likes on-leash hiking, you can also check out their once-a-month "Paws on the Path" hikes... my dog loves them, and it's a good way to learn about some of the dog-friendly trails on Oahu. You can find info about upcoming hikes here
Dog Hiking | Hawaiian Humane Society (http://www.hawaiianhumane.org/dog_hiking.html - broken link))
Many but not all beaches are dog friendly, and though there are technically leash laws, there are lots of places where well-behaved dogs romp off-leash in the sand, as long as they aren't bothering anyone. (I like to take my dog to the less-crowded end of Kailua beach, and mornings are the best time for fewer people and more dogs.)
It's hard to give specific rec's unless you know where you're going to be living, though. If you're in Ewa, you probably won't want to drive to Diamond Head for their off-leash park, for example. Once you get here and walk your dog around the neighborhood, you'll meet neighbors who have dogs, and they'll give you the local information.