One common type of activity is attempting to make direct contact with a distant station. This could be using a voice mode, digital mode (text), or CW, which just means using Morse code and a keyer to communicate.
Operators can make contacts all over the world using the Earth's ionosphere as a mirror, to reflect signals, and send them further than any method of terrestrial propagation can achieve. This is called skywave propagation as opposed to groundwave propagation, which is like how cell phones work.
The caveat being that skywave propagation appears to have a degree of randomness to it. It isn't totally random but we collectively do not understand it enough to fully predict when and where our signals will end up.
So there's an opportunity to make direct contact with another station somewhere else on Earth, depending on the propagation conditions.
So to now actually answer your question, there's novelty in getting your station working, and then using it to have a brief conversation with a random person at least hundreds and sometimes thousands of miles away. This is known as a QSO.
At different times, for example, with my station being located in North Carolina, I might suddenly be able to reach stations in Europe around sunset. Usually, it's more limited to random areas within the United States with a basic antenna, but not always.
If I announce my presence over the radio (call CQ), and someone comes back to me, we'll exchange some basic signal quality info. Afterwards, if the other person wants, you can ask about their radio setup, and talk about common interests, while avoiding some topics like local politics, etc.
Here's a video of someone doing this: https://youtu.be/tXIbSGX2SvU?t=3m40s
That's only one activity while there are many others. Another activity would be pointing a beam antenna at low earth orbit amateur radio satellites to make as many quick contacts with other operators as possible while the satellite is above the horizon. This is a very physically and mentally challenging activity. Here's a video of that: https://youtu.be/QWtdyT4PYMw
There are also digital modes as well. FT8 is a common digital mode for making contacts with much less personal interaction using a PC. This one is easier to just watch and see than to explain: https://youtu.be/bSpyYV-xz5c
Hopefully that gives you some ideas about what it's like to make contacts over the radio. Some people like the social aspects and others (like me) like the technical and experimental aspects of making contacts and seeing where and how far my signals will go.