“When a word have like 5 meanings, it feels like I'm learning 5 new different words.”
Often the meanings are related, but when they’re different, I agree with you - it is like learning five words. In that case, it is useful to study them as five separate words, learning the meanings in order of usefulness to you. Miriam says how to do this in her comment, and I agree wholeheartedly. It's not clear how she got downvoted, but here’s a thumbs-up from me.
By the way, Neil, I find sentence structure and pronunciation to be key for my Brazilian students learning English. As the English has a lot of Latin vocabulary, lexicon is usually the easiest part.