Attempting to study every single word and have complete understanding of everything that’s said is generally more trouble than it’s worth. Especially with native media, you have an opportunity to directly access understanding of a word or phrase or conjugation through seeing how it’s used by natives. Imo that’s far more valuable than looking up a general English definition of a word and trying to grasp the meaning of a Japanese word by proxy through English. Plus, there will be lots of context-specific uses of words or phrases that don’t really capture a word’s meaning or use all on their own. It will take a lot of exposure to words like that in order to actually develop understanding of their meaning and usage.
Consider a phrase like おつかれさまです. You can translate it into English, but there’s really no culturally equivalent meaningful analogue, so if you want to understand it, it’s best to simply see how native speakers use it and mimic that until you have your own internalized intuition for it. Incidentally, that’s generally how people acquire their native language; from watching other native speakers use it.
I recommend watching an episode or two and studying words that are repeated a lot, or seem particularly interesting or useful to you, then periodically going back and rewatching earlier episodes after having studied words from them for a while. I would not recommend waiting until you’ve studied every word from an episode to move onto the next episode; that would drastically slow your progress and probably make watching the show much less interesting too.
It’s also important to note that seeing more examples of a word in context is far more effective study than reviewing a flashcard for that word a hundred times. You might study 忙しい and be able to draw up the word “busy” when you see it on a flashcard, but your understanding of and ability to remember and use 忙しい will be vastly better if you watch a dozen examples in context of someone saying ごめん、忙しいけど, or 今忙しくない? or うわ~、今日めっちゃ忙しそう! because our brains require connections to access memories, and words understood in context are far more connected than compartmentalized vocabulary study. So while actively reviewing individual words is still useful, it’s much better to review a little and consume a lot.