At least you gave us enough notice that we could respond!
Where in Panama are you going?
I've been diving the Caribbean side for a number of years at both Bocas del Toro and Portobelo. In Bocas, I recommend Bocas Water Sports, a 5-star PADI operation with excellent DMs. Here's a review from two years ago:
www.scubaboard.com/forums/central-america/323572-back-bocas.html. What's interesting is that BWS has some regular customers who come there from Costa Rica because, they say, the diving is much better.
The only downside of Bocas is that it's a 10 to 12 hour drive or a hour flight from Panama City.
Portobelo is much closer to Panama City; just 90 minutes by car. There, I recommend Scubapanama. Panama Divers is another operator there, but they charge much more per dive and for lodging. Both operators go to the same dive sites, but Scubapanama is much closer to them. That may not sound like a big deal, but in a small, open boat bouncing over the waves and being blasted by one of Panama's typical rainstorms after a dive, I'll go for the shorter distance any day.
Scubapanama has clean but rustic cabins, too. They're also much closer to the town of Portobelo, which is well worth seeing. And, it's close -- very close -- to a couple of good restaurants.
On the Pacific side, there's Coiba, a large island and ex-penal colony surrounded with great diving. Coiba has a reputation for world class diving and a great opportunity to see large schools of barracuda, tuna and sharks of all types. It's also much more of an advanced dive -- blue water diving, strong currents and surges at times. I haven't dove it yet because getting there (and back) from Panama (the city) takes a large chunk of time. It's an 8+ hour drive from Panama to Santa Catalina. From there, it's from 1 to 3 hours to the dive sites in small boats. The Pacific is also a 3 to 5 mm wetsuit versus the Caribbean where I dive with just a skin and my wife wears a 3 mm. There are a few dive operators that work out of Santa Catalina. You should check the past posts here on Coiba if this interests you.
Also on the Pacific are other locations: the Pearl Islands and Taboga and Pedasi on the Azuero Penninsula. Taboga and Las Perlas are much closer to Panama, but very expensive and, from what I understand from my Panamanian diving friends, not a lot to see. My friends also said Pedasi is disappointing due to its poor visibility and not a lot to see.
Let me know if you need any more info -- travel tips, driving directions, etc. My wife and I will also be going back to Panama, in April. We'll probably be diving Bocas.