What dialect of Spanish do the locals speak in Panama City?

Spanish is the common language in Panama and is not much different than Spanish spoken in the rest of Central America. Even if you do not know the right Spanish, you will be understood, although you will find that some words are local.

While many people understand English, those are mostly concentrated around tourist areas. Although not everyone will be bilingual, there is often staff that speaks English in international hotel chains. As in most places, how much you manage without the language depends on who you interact with and what you are trying to say. If you learn a few Spanish words from a phrasebook, you can intermix to get what you are saying across.

I'll answer it a bit differently, since you asked about English being popular as well.

Officially, in Panama (the country), 93% of the country speaks Spanish as their first language, although many speak it as their second as well.

English is spoken by 8% of the country.

So by and large, as a traveller, you'll get by very well if you speak Spanish and/or English. Indeed in large cities, English tends to be more common than in the country, so Panama City will have more than 8% English speakers.

In terms of what Spanish, it'll be Panamanian Spanish. It's basically the same as European and other Central American Spanish, but with words borrowed from Italians, Greeks, East Indians and Chinese. This page also demonstrates the large variety of Panamanian Spanish slang if interested.

Just Spanish bro, just spanish but English also used but less, The index of linguistic fractionalization in Panama is 0.4 so it's meant not only spanish is possible ;) So maybe it's will be useful for you !