There's stuff to unpack in there. How many countries do you think there are? That depends on what the meaning of country is, right? For American hams, the most commonly recognized source of authority on that is the American Radio Relay League, who likes to call themselves the National Organization for Amateur Radio (which is more recognizable to non-hams). The ARRL manages this through their DX Century Club programs. (Note that to Shortwave Listeners and other radio hobbyists, the abbreviation DX has long been used to denote Distance and the Unknown). Part of the rules are common sense: a country is a place under its own government (even puppet countries, like the old Soviet Republics, were officially their own governments). Where the DXCC list deviates from that common sense is that if one part of the country is separated from the rest by more than 350 kilometers (217 miles), that remote portion is considered a new country. That means, for example, that Hawaii and Alaska are states in the United States, but they're also separate countries for the DXCC awards!
That means that there are more ham radio countries than countries recognized by other governments. There are 340 currently recognized DXCC countries. That includes countries that don't exist anymore, but if you have a valid confirmation (QSL) card from the country, it's still accepted. For example, at one point, the Panama Canal Zone was a country - I have a card from back then.
That's just the very edge of that rabbit hole, but I won't go into more. Suffice it to say that whenever there are lists and recognition that comes from checking off having contacted ("worked") these countries, guys get competitive and go to great lengths to work more countries than their friends. Lists like the Most Wanted Countries circulate, and obscure, uninhabited islands attract “DXPeditions” to get operators on them for brief periods (typically two weeks). This was a major operation in 2006, Peter the First Island. As life works out, I know one of the operators in the group casually - he works in town at one of my previous employers
One of the areas of the radio hobby you'll need to study is radio propagation which tells you how radio signals get from one part of the earth to another. I've posted on this topic before; this post gives a good summary of the essences and even some of the details of current ham operating.
Probably the thing I find the most interesting in radio is propagation - how the signal gets from one station to the other - and especially the ionospheric propagation. This can turn into multiple pages itself, but the ionosphere is a layer of the atmosphere where the molecules present in the lower atmosphere are ionized by incoming solar radiation, and the air is so rarefied that it takes long times for ions to bump into something that makes them neutral again. ("Do you think you're ionized?" "Yes, I'm positive") . The ionosphere, in turn, is characterized as having several layers, with each layer's name changing with height. The lowest, densest layer is called the D layer, and as we look farther vertically, they go through the E and F layers. During periods of high ionization the F layer can further stratify into F1 and higher F2 layers. The ionosphere expands and contracts, getting taller or shorter with incoming solar energy.You rarely have the opportunity or need to talk with the exact opposite side of the world. That's called the antipode and many times, there's nothing there but ocean. The complication of contacting a station even near the antipode is that (often) the radio signals from each station diverge a bit from a straight line, and arrive at the receiver after varying time delays. This multipath propagation can distort the modulation and make it difficult to understand the transmissions.
In general, the higher layers are active when the sun is overhead, but even then are dependent on the solar energy output which varies day by day, with the solar cycle, with (possible) grand cycles of solar activities and so on. Generally, the type of propagation that gets hams excited is from the F2 layer, for a simple reason: it's the highest layer, so signals "reflect" from farther up and can go farther around the world.
This map from Engaging Data shows a set of crosshairs that are antipodes of each other. On the left, I dragged the globe around until it was relatively close to my location, and it repositioned the globe on the right to show the antipode. When we were kids, we used to say you'd dig straight through the earth and come out in China. Most of the US has its antipode in the South Indian Ocean; China is in the Northern Hemisphere.
It works out that one of the handful of countries I haven't worked that are still on the list is an island near Indonesia in the top of that view. Called by the awkward name Cocos-Keeling Island, it's an Australian possession and has an Australian callsign - starting with VK9C. In this case, it's VK9CZ. Unlike the Peter 1 expedition, which looks like a military takeover of the frozen island, it's two guys on a two or three week trip to the island, which is a lightly inhabited resort.
I don't consider Indonesia particularly hard to contact, but that was when we had sunspots* and I'd work them on 15 or 10 meter bands in our evening local time - early morning there. One of the first steps is to try to get a competent propagation prediction. The VK9CZ guys provide access to customized maps for your location, provided by this site (I believe). I get this:
You supplement these predictions with the presence of GrayLine propagation enhancement (most important on 7 MHz and down and vitally important on 1.8 MHz (160meters)) which happens at local sunrise and sunset plus or minus a bit - perhaps as much as an hour. The GrayLine would add higher probability colors perhaps red or orange on the bottom row (80 meters), at 12 and 19 horizontally.
This prediction shows that 15meters will have great propagation to VK9C at 15 or 16 hours UTC, the deep orange or red squares. That's 10 or 11 AM here and makes some sense - except for the fact that they're almost perfectly 12 hours out from us, making that well after dark on their end and those are daylight bands. So far this particular propagation prediction chart has been pretty useless. This morning I was up at 6AM for the sunrise GrayLine. They were on higher frequency bands but since they're having a really hard time working the US, they dropped to 80 meters. I listened from about our sunrise at 6:50 until about 7:30 and never heard any signal I could identify.
I need some more study of the band openings. Mostly, I need some luck here.
* I have plenty of posts that mention sunspots and how higher sunspot numbers correlate with higher solar activity and better propagation, but nothing on that topic by itself.