The big question is, how long are you going to be in Panama? Coiba, in the Pacific, is supposed to be world class diving ... lots of exciting big stuff. The catch is, once you get to Panama City, it will take a good day to get to Santa Catalina. From there, you're still hours and another day by boat to Isla Coiba. If this interests you, check out the Coiba Dive Center online. A word of caution: I haven't dove Coiba, so I can't give you first hand experience, but I have been told by dive instructors who have been to Coiba that it is more advanced diving. The depths are greater and there are often very strong currents to contend with.
My favorite diving spots are Portobelo, with ScubaPanama, and Bocas del Toro, with Bocas Water Sports. Both are on the Caribbean. Bocas del Toro is near the Costa Rican border, and about 90 minutes by air (via Aeroperlas or Air Panama) or a very long day's drive (10+ hours) and a half-hour water taxi ride from Panama City. With the new cross-isthmus toll road, Portobelo is about 90 minutes from Panama City.
Don't expect great visibility in the Caribbean. 30 to 40 feet is about it. But, both Bocas and Portobelo have magnificent varieties of coral, fish and other critters. On my last dive in Portobelo, I was actually followed by two usually very shy French Angelfish.
Both towns have a lot to offer out of the water, too. Bocas is a surfer's and backpacker's mecca. It's a town very unique to Panama in the rest of Central America. When you're not diving, the snorkeling is among the best in the world. The snorkeling reefs are in 4 to 6 feet of water, so everything is close-up and clear.
Portobelo is steeped in history. You can walk through the ruins of 15th century Spanish forts that were sacked by the pirate Henry Morgan. You can walk through the Customs House which once held 1/3 of the world's wealth in Inca gold and silver. And, you'll be diving off one of the most picturesque horseshoe shaped, palm lined, white sand coves near the spot where Sir Francis Drake was buried at sea. Maybe you'll be the lucky one to find his lead-lined coffin.
Check out the websites for Bocas Water Sports and Scubapanama and then take your pick. I don't think you'll be disappointed.