Internet in Panama
There is good internet coverage in most parts of Panama, with wifi available in hotels and apartments, as well as free access in malls and some other public places. To get wifi installed in a house, apartment or office, just choose a supplier and take your ID and proof of address. Phone companies can also give you mobile data, so you can be in touch with your email etc. virtually anywhere.
Quality and speeds vary but are generally sufficient for satisfactory Skype video calls and so on.
While it might seem that the whole world now has internet access and if somewhere doesn't, it should have, the places that don't are probably run by people who don't use it much themselves. Or they may just be saving money. Either way, it pays to be self-sufficient and if you have to rely on someone else's system, try it out first to make sure it works.
While internet cafes are fast disappearing in communities where everyone has smartphones and home wifi, in less developed areas - even in quite small towns - they still exist. It might be somebody's front room or the corner of a shop, and the equipment might be rattly old keyboards and bulky old computers, but when you need to get something done, the presence of the equipment is more important than its age and condition. If it works, that's all you need.
Useful links: