I can most definitely help you out, since I am currently in Panama.
8 days in Panama, here's what I would do:
- Go to the Panama Canal, watch the ships go by, very interesting (http://www.pancanal.com/eng/anuncios/cvm/index.html)
- Go to Casco Viejo, the old quarter of the city which is very quaint and interesting, lots of restaurants and a couple of bars there, lots to see (http://www.cascoviejo.org/content/cnPanels.php?cid=10)
- Maybe have dinner at the Causet Way, in the city, it's nice out there and there are a number of dining options (http://www.amadorcausewaypanama.com/)
- Take two days and one night (or three days and two nights) to go to San Blas, the beautiful archipelago in the Caribbean just a short flight away (http://www.sanblaspanamahotels.com/)
- Take the Panama Canal Railway, it's a great day to travel alongside the canal (http://www.panarail.com/home.html)
Those are the highlights, IMO. I would also stay in the city; there are lots of different options and this way you will be close to the different attractions and points of interest. There is a Marriott Courtyard on the Cause Way which is right on the Canal, that might be a good place to stay at.
Do not take any buses! They are dirty, messy, and sometimes dangerous. Take taxis everywhere and beware: they do not have meters. So be prepared to be charged ridiculous amounts if it is obvious you are a tourist; what you can do is ask around before going somewhere so people can help you determine the approximate cost.
As with most places in Latin America, take some basic precautions when heading out: don't carry too much cash, pay attention to your surroundings, etc.
Good luck! Hope you get good service, which is scarce in Panama.