I have done a lot of international flying, in private light aircraft.
In Europe/USA/South Africa/Australia and a few others one files a flight plan (sometimes a day or so before the flight) and goes. The sign of a civilised country (IMHO).
Outside these places it gets more complicated, previous permission needs to be obtained sometimes weeks ahead. Each country publishes a document called the "AIP" or "Air Pilot" which includes, amongst a lot of other stuff, what they want to know and who to ask. It gets more difficult when the relevant person never answers your request, in which case you have the choice of aborting the trip or just turning up and hoping for the best.
The other way of solving this problem is to use a clearance agent who often has agents in country who know who to ask and/or who's palm to grease to get quick permissions. This service costs of course.