Yeah, my Star Trek knowledge starts and ends with the movies (both the OG Trilogy and the two Abrams films), so most of these meme might as well be in Klingon. But everyone else is having so much fun that it's fun laugh along like I understand the jokes.
On a related note, if I wanted to get onto Star Trek, what series does everyone recommend starting with?
The newer series do fan service referencing old series so you may miss jokes and references. Anyway, Strange New Worlds would be a good starting point. It doesn't matter if you miss references to enjoy it.
Older series "require" less previous knowledge because there is less lore to base them on. But sometimes the stories... didn't age well. Most of the episodes did, but for example the 1st season of TNG can be hard to watch for younger people.