What's your budget? Panama is pretty full at the moment and short-term apartments are difficult to find, but I'm sure these sites can help you find something:
http://www.panamaforum.com/living-panama/16-renting-apartment-panama-city.html
http://www.panamaforum.com/living-panama/1307-anyone-has-apartment-rent-october.html
Also, I noticed you're planning on staying in Panama for 60 days. Are you aware that US citizens can only stay for 30 days? You can get an extension while in Panama, but it's pretty complicated, you need proof of funds, a Panamanian to vouch for you, and go to the immigration office several times. It might be easier to fly to Costa Rica or Colombia (or Ecuador, Haiti, Dominican Republic, ...) for 48 hours and then return for another 30 days.
The food in Panama is okay, the supermarkets are pretty good, and there are several vegetarian restaurants but they all use way too much fat for my taste. Do you like Mexican food? There are a few pretty good Mexican restaurants and they have a lot of vegetarian food. Also check out the Organica health food store in the Paitilla Mall; it's not that big but still the best in Panama. (Also try some green coconuts.)
Enjoy your trip... and I hope you donate to RP tomorrow! :)