Lurk, as other people have said. You can figure out which shows to watch just by seeing what people talk about and reference. Reverse image searching character images, reading file names of screenshots, paying attention to name drops and googling those, etc. These are all strategies I used when I first started lurking /a/ in order to figure out what people were talking about. It's also a good way to find fun and interesting anime to watch.
Don't feel like you have to watch everything though, I sure didn't. In a lot of cases, some stuff is just posted so frequently you learn about it through osmosis, you don't even have to watch the anime to get a reference to it. Evangelion is one show that's still true for. K-On and Lucky Star were that way for a while. Madoka too. Monogatari series as well. I actually bothered to watch these series eventually, but by the time I did I already knew a lot of the character names and had a general impression of the show, just due to all the discussion I saw on /a/.
Some other series I picked up due to seeing them talked about on /a/ frequently include: To aru Majutsu no Index (light novel, anime) as well as its spinoffs, Highschool DxD (anime, then light novel), Hunter x Hunter (manga), Fate/Stay Night (visual novel) as well as all its spinoffs, Yuru Yuri, Saki, Vinland Saga (manga), Legend of the Galactic Heroes, and a lot more.
Lastly, in terms of general culture, it's good if you have a broad knowledge of anime, rather than narrow. Being able to make inferences and connections between series adds value to discussions, not in the sense of "X is just a ripoff of Y" but being able to see influences of other series and recognize patterns. It allows you to construct jokes and to understand the jokes other people make. This pic I posted is an excellent example of the kinds of references and jokes people on /a/ make due to broad anime knowledge. Once you can instantly get stuff like this, you'll have no trouble blending in.
