It really depends on what you want to do/see during your visit and your mode of transportation. As for the weather, it should be comparable to what you experienced in Costa Rica.
If you are interested in local culture, I can recommend a few places you may find very interesting. I can not comment on budget accommodations in Panama City as we chose to "splurge" for our first and last two nights there - staying at the Hotel Continental Riande in the financial district. I can also say stay away from the Hotel Isla Grande on Isla Grande. It's cheap but far from clean and the garbage dump is just on the other side of the very high chain-link and barbed-wire fence. Though the vultures love it.
My recommendations:
Santa Clara had a very nice beach (Playa Santa Clara) and several places offer kayaking tours from April-November.
Penonome is on the beaten path on your way to other places. The Hotel Guacamaya is the best in town (clean, hot showers, private rooms, etc.) and ran about $22USD/night. They also have an interesting Chinese restaurant next door where the tour buses stop for breakfast and dinner. (We had fun watching a bus load of traveling retirees over breakfast. The food was very good too.)
El Valle de Anton is a very cool community built in the caldera of a volcano. It has a very large "farmer's market", canopy tours (~$45/person). It's an interesting place and because of the location, it has a very interesting weather pattern. There's a strong wind for 10 minutes, then 10 minutes of calm - you just hang onto your beer and don't talk during the wind. We stayed at Hotel y Restaurante Los Capitanes which is owned by a German ex-merchant marine captain. He loves sitting with the guests and telling stories. He will also introduce you to everyone else over meals. The fare is German, of course, and quite good but there are several local eateries available too.
Santa Fe is a very small village with lots of flavor. It is home to the "Orchid Lady" (Bertha) who has the largest collection of local specimens around. She is known around the world for collection. There are also several waterfalls and access to the rivers for tubing and swimming. Horseback riding is available too. There is only one hotel in town - Hotel Santa Fe. It has 21 rooms, all brightly decorated and cozy. Each has a private bath but cold water showers only. (Actually felt good after a day in the heat.) The rooms were very clean. The small restaurant specializes in the fresh fish caught that day. It's a great place for meeting other travelers.
San Francisco is on the way to Santa Fe. There you will find Inglasia San Francisco de Veraguas - a church dating back to the 1700s. It contains a unique mix of christian and pagan symbolism throughout and was renovated about 5 years ago to bring more visitors to the village. There is also the Chorro del Spiritu Santo which great for a swim by the waterfall.
Boquete is located in the mountains and is very pretty. There is a growing number of American ex-pats moving there but they tend to stay in their gated communities and the town isn't so modern that it's lost it's charm. It is elevated enough that there is a constant mist from the clouds passing by but you never get wet. The vapor evaporates on your skin immediately. We stayed at the Hotel Panamonte which ran about $65USD/night - not budget. There are several budget accommodations available in town.
If you are choosing to use the local buses for transportation, the starting point and end point are on the front of the bus. "Panama - Boquete" means Panama City to Boquete with stops in between. Be sure to ask the driver where they stop because not all buses making that trip stop at the same places.
Enjoy planning your trip!