Global Tuners is a worldwide project for people to share their shortwave radio receivers. There are about 50 tuners around the world, and once you create an account you can log in to each of them, change the tuning settings, and listen to the broadcasts that can be heard in that part of the world.
I have an old shortwave set that was my stepdad's, and so I've been trying to pull in stations from home. That's been successful up to a point, but compared to the last time I was following this world (20 years ago) there's a lot less in the English language and more in Chinese. Getting access to tuners around the world, especially to ones that have better antenna setups than I do, is remarkably handy.
The other thing that you can use a shortwave set for most of the time is listening to AM and FM stations that are aimed at a local audience. Here is where some of the most unusual listening comes in - it's just plain fun to tune in an Australian FM station (via a tuner in Australia) and listen to what ordinary radio sounds like there.
For best results, you'll want to pair this with some lookup tools that help you figure out what's on the air at that time that's likely to be heard in that part of the world. Short-wave.info has a nice real time database of broadcast schedules, so that if you are tuning around the dial on shortwave you can see which stations are likely to be broadcasting. FMSCAN has global listings of AM and FM stations, and MWLIST has longwave, mediumwave, tropical bands and a shortwave radio database. It's the kind of information that I recall from sorting through the back pages of the World Radio and TV Handbook, except that it's all linked together (and more up to date than my 1980s copy).