I went to Panama City on a trip with my high school in 2009, and absolutely loved it. We mainly stayed in the city areas, but we visited these places:
Albrook Mall (this mall was huge! we made about two trips here! loads of great priced items)
Gamboa Rainforest Resort - this was one of the highlights of the trip! I would definitely recommend this to you. The resort is HUGE, and has an amazing view of the local rainforests. We walked around and there were bird aviaries and butterfly conservatories, and there were two cable cars that take you to the top (great view!) and then back down. I even saw a sloth sleeping in the trees! There was an amazing restaurant, and while we were there, there was rain (it lasted for about half hour) and we saw lightning flash over the tree tops - spectacular!
Avalon water park - this was fun, and had lots of water slides and pools. It's a family place, and you get get things like burgers, fries, hot dogs etc.
Panama Canal at Miraflores - this was another highlight. We saw an actual cargo ship come through the Canal! we also had lunch there. Miraflores is the name of one of the lochs. We also went into the actual canal on a boat trip!
There was a museum in there, and there was a section with photos of the Canal being built and pieces of equipment from that time. My great grandfather was a worker at the Panama canal, so this was very special for me.
Party Bus! It's a school bus madeover into a party bus. It was painted black with glow in the dark designs, and inside had a bar, DJ area, sound system and seats! We had it rented out specifically for our group, but you can go on one with a smaller group and meet other people.
Overall, I loved Panama City, and I want to go again!