i’ve gotten the reading for each of them down, but I’m having an extremely hard time remembering which means which. Is there a mnemonic/general way to help me remember their meanings because, honestly, I’m getting tired of those three being the only ones I’m getting consistently wrong.
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For 大きい – 大きさ it’s easy:
All い adjectives turn into nouns by replacing い with さ
大きい (big) – 大きさ (size)
暑い (hot) – 暑さ (heat)
and so on.
As for 大した、I recommend just memorizing it 
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大きい is the main one I guess. 大きさ is a regular conjugation of い adjectives to turn them into nouns, so it necessarily has the same reading as 大きい. See さ (JLPT N4) | Bunpro for the general grammar point. So 大きい is “big” and 大きさ is “bigness”, aka size.
大した originally comes from 大 + した (past form of する). It’s not uncommon for single kanji to be read with the onyomi when paired with する or the related form じる. Consider 信じる、感じる、命じる、関する、対する…
I suppose you can think of the meaning of this one as “did big” (like, it was a big hit or something). If it “did big” then it’s “great”.
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大きさ → this one means ‘size’ - you can tell because the final さ character is pronounced the same way as the start of size! Like さize!
(whoops, replied to the wrong person initially!)
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I think the others explained 大きい and 大きさ well enough.
For 大した it helped me a lot to know the phrase “大したことない” “No big deal”.
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I read. That’s how.
Don’t worry too much about it, but generally 大きい and 大きさ are just literally talking about sizes, where as 大した is like “greatly” or on a great scaled or considerably (large in scope). It’s easier to just think of 大した as it’s own separate thing while the others are physical size based for now.
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This will get easier especially as you study grammar. Plain memorizing which syllables put together make which meaning is very hard, but when you know why that’s the case (for instance, replacing the final い with さ on い adjectives makes it the noun form, kind of like adding -ness to an adjective in English, and this works with other adjectives too), it becomes much easier.
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