This is a quick map of places I've been. It doesn't include Antarctica, but since that's not a country per se I can hardly complain. It's probably also wrong in the Caribbean and Southeast Asia.
Create your own visited country map. Try this; it's a lot of fun if you've been anywhere. The site also has similar maps for the US states and the Canadian provinces and territories (I've been to all of the states and to all of the provinces, but none of the territories).
Here's my husband's map; he went to Finland, Ukraine, and Russia before we were married. Actually, he went to the USSR, which then included a lot of states that are independent now, but I didn't include them.
My husband and I decided to travel when we were young, rather than waiting until we'd retired. When we arrived at the airport in Cairo, Egypt, in 1978 we saw a group of older Americans who were waiting to depart for home. Frankly, they looked dreadful. Exhausted, sick, overwhelmed. We decided then and there not to wait to see the world.
NASA sent us both on a number of overseas trips, mostly to NATO or AGARD conferences, but sometimes just to visit aerospace research organizations. For the latter, we mostly took a few days from our own pleasure trips, for which we, not NASA, paid our airfares and expenses. We just saved a day or two of annual leave and had all the security, etc, formalities taken care of by the government. Generally, it's impossible for a foreign national to just drop in on such establishments; invitations have to go through embassies. An example is the CSIRO research establishment in Melbourne, Australia.
In 1991 we spent a month in Australia and a day there, visiting and talking about aerospace research, mostly in relation to F/A-18s, which the RAAF flies. They sent one of their young engineers to Dryden on an eighteen-month exchange tour about two years later. He worked on the F-18 HARV, High-Alpha Research Vehicle, doing stability and control parameter estimation. While he was there, I got him a flight in the variable-stability Learjet, just as I did for Dryden research engineers, to show him what aircraft handling qualities feel like. I also wrangled him a ride in the backseat of an F-18 just before he left. That was a real thrill for him. His exchange tour would probably never had happened if we hadn't stopped by that day and talked with his management about it.