According to Seafood Watch, branzino is listed as a "Best Choice." The fish are small (about 1 pound each), so it it make the most sense to cook them whole. The bones add a richer flavor and keeping the fish whole help prevent it from drying out. 

Grilling and roasting are the best ways to cook whole fish. Here is a super-easy way to prepare it.

1 whole branzino (about 1 pound)
Salt and freshly ground pepper
2 thyme sprigs
1 lemon
1 tablespoon extra-virgin olive oil

Light grill, wait about 15 minutes until it is really hot.

Using a sharp knife, make shallow diagonal cuts in the skin every two inches or so from the dorsal fin to the tail. Cut 3 thin slices from the lemon, cut the remaining piece into wedges. Sprinkle salt and pepper in the cavity, and insert the thyme and lemon slices. Rub the outside of the fish with olive oil and season with salt and pepper.

Generously oil grill grates. Grill fish for about 5 - 7 minutes per side. Serve with lemon wedges.

Serves 1 to 2 people (depending on size).

For information on how to fillet cooked fish, a great resource is the Bon Appétit article How to Fillet and Serve A Whole Fish

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