I haven't posted much on this forum, but as my questions were not that different than most other people travelling to Panama, I managed to mine almost all of my answers out of other posts (the search function really does work!). The people of Panama are very proud of their country (moreso than most countries I have visited) and are very friendly if you engage them about their nation. Short highlights:
Panama City: We stayed in Casco Viejo - I don't know why anyone would stay outside of Casco. While the area is quickly being gentrified, there are still authentic snipits of what life looks like in Panama. The thing I like most about emerging areas like Casco is that people aren't afraid to be innovative, whether it be in restaurants, or shops or architecture. We ate really well. with a couple of misses, but highlights for us were Tantalo Kitchen, Veggie Moon (fantastic seafood) and Capital Grill (amazing views and very innovative cuisine). We spent time at the usual tourist haunts (Miraflores Locks, Gamboa area boat tour and a Canal Transit), but Casco was the unexpected gem for us.
Bocas del Toro: Real end of the line, frontier backpacker vibe, with lots of interesting people. We spent one night in Bocas Town (great food at El Utimo Refugio) and three nights on Isla Bastimentos "off the grid". Bocas is beautiful, with great snorkeling and a laid back vibe, but not for people that are used to a 5 star island experience. We really liked it, but I can see why it would not be for everyone. I don't think there are too many places left in the world where you can be secluded in such a tropical setting.
All in all, it was a great trip - I would recommend visiting Panama if you want to have an adventure, not just a laid back vacation...