Just keep an eye on your left-hand panel during supercruise, and you'll see listening posts and other odd POIs from time to time. Note that these things only appear in your panel when you're less than 1000ls away, so you'll have to fly around a system a bit if you want to verify that there is *not* anything around.
I also recommend reading the *local* papers at various stations. Some of these contain stories which hint at mysterious things in certain systems or planets, and they will provide a starting point for your searches. I'm not 100% positive but I think these stories might be exclusive to papers with the word "herald" in the title.
edit: I got into the listening post clue hunts entirely from in-game, because I really detest the "gaming via YouTube style of play that Elite seems to be built around". But yeah anyway the listening post stuff is somewhat signposted within the game. I will warn you that some of these clue-hunts are dead ends, or finish off by pointing you to a planet where there's something interesting, but it gives you no coordinates. Also you'll in some cases have to grab a pencil and paper to decrypt codes, etc; and I'm sorry to say that some of the encoded messages and some of the decryption keys contain typos, which can make it waaaay harder to figure out.