Quoted from https://quizlet.com/41134323/hongdae-grammar-prep-1-flash-cards/ *그런데 But/However/And If you want to contrast two sentences and say "A + but + B" you can choose to use either 그렇지만 or 그런데. If you want to introduce two actions or states that occurred one after another, and if the first sentence works as background information for the second sentence, only use 그런데. 어제 이거 샀어요. 그런데 정말 커요 그런데 means "but" but at the same time it can mean "and" as well. If what the speaker is implying is the meaning of "and", the entire sentence can mean "I bought this yesterday, and it's really big." or "I bought this yesterday, and as I found out, it's really big  *그렇지만 But/However If you want to contrast two sentences and say "A + but + B" you can choose to use either 그렇지만 or 그런데. 어제 이거 샀어요. 그렇지만 정말 커요 그렇지만 means "but" or "however" so the speaker is contrasting the two facts- "having bought this yesterday" and "this being too big" - so it sounds like the speaker is disappointed that it's very big. *그래도 "but still", "however", or "nonetheless" The literal meaning of "그래 + 도" is "even if you do that" "even if that happens" or "if you do that too" and the meaning of "still" is added to the context. 노래방에 가야 돼요. 그래도 노래 안 할 거예요. = I have to go to a 노래방. But still, I'm not going to sing. 요즘 바빠요. 그래도 한국어를 공부하고 있어요. = I'm busy these days. But I'm still studying Korean. As you can see, 그런데 can be used for a wider variety of meanings, and in addition to that, 그렇지만 has a very formal nuance to it, so in actual everyday conversations, 그런데 is much more commonly used than 그렇지만, and 그렇지만 is more commonly used in written language.

To simply put, 그렇지만 means 'but' or 'however', 그런데 means 'by the way', and 그래도 means 'even though', 'still', or 'in spite of'. Examples) 네 말도 일리는 있다. 그렇지만 우리는 다른 사람들의 의견에 따라야만 한다. (You use 그렇지만 to say something contradictory to the statement that has been said) 아 그렇군요. 그런데 왜 그때는 말씀을 안하셨습니까? (You use 그런데 when you change the subject of conversation or want it to lead in a different direction) 소리를 높여 다시 불렀다. 그래도 대답이 없었다. (You use 그래도 when you tried to change a situation but the result remains the same) Sometimes they are interchangeable and do not make much difference. The best way to understand the meaning of each and the usages would be to know them through the context they appear. Hope this helps.

Thank you guys for the great answers, they helped a lot! 감사합니다!