There is a well-established method of doing this, shortwave radio, also known as "High Frequency" or HF. Radio waves in the frequency range from 3MHz to 30MHz (10 to 100 metres wavelength) can be reflected from the "ionosphere", a layer of electrically charged atoms high in the earth's atmosphere. They can also be reflected off the earth's surface, including the oceans. This allows shortwave transmissions to reach anywhere in the world via a series of reflections back and forth between the earth and the ionosphere.
This isn't used for everyday broadcasting for several reasons, including:
It can reach anywhere in the world, which means another radio station using the same frequency anywhere in the world can interfere with it. Most kinds of radio broadcasting are only relevant to a limited area of the world, because of language and political boundaries, so using methods that can reach everywhere isn't useful.
The quality of speech and music is variable, but tends to be low. You're relying on the atmosphere to behave in the way you want, and it takes no notice of human concerns. Shortwave transmissions tend to work better at night, but that's not reliable. The earth's magnetic field, the solar wind, sunspots, and likely other things can affect shortwave, and there isn't much you can do about them. Switching to a different frequency can help, but this is not reliable, and gives your listeners a new problem, finding your station.
Shortwave is used for applications where you need long-distance communication without using infrastructure, such as undersea cables or communications satellites. This includes backup military communications, backup long-distance communication from ships and aircraft, international propaganda, spies being sent messages by their controllers ("numbers stations"), specialised kinds of radar, and amateur radio enthusiasts.
It's fascinating to get hold of a shortwave receiver and tune through the frequencies. You'll find a weird world of crackly speech in a range of languages, plus strange noises. The BBC World Service broadcasts on shortwave, as do many other national stations. Shortwave-capable receivers are available from any good electronics store.
This is the obvious and correct method for trying to find out if there's anyone else alive in the world. If you want to transmit, you need a radio amateur's equipment, and in a world where most people have vanished, the easy way to find that is to tour a city looking for a house with unusual antennas. Learning to use the gear is necessary, but an amateur will usually have manuals for their equipment, and more general books.
To research this properly, locate your local radio amateur, and explain what you're writing. They'll give you far more information than you need.