I’m not sure where you are in the series so I guess I’ll try to avoid talking directly about stuff that happens in later books (I’ve already read them all so it’s a little hard). But SPOILER ALERT for the show up to Season 2 at least.
Tyrion Lannister - I liked him from right from the beginning. He seems to be the only person in the whole series who has a brain and also has a bit of a conscience too. He’s probably the most popular character because he gets all the good lines and even though he plays on the side of the “bad guys”, you don’t really see him as a bad guy because he does acknowledge that his family is kind of evil. At the same time, even though he’s fun and witty and definitely the most sympathetic Lannister, he does also hold the same sexist attitudes that men in that world hold. I do feel like his character stalls later on in the series. By A Dance with Dragons, I’m just not sure where he’s headed or what his purpose is.
Catelyn Stark - She gets a lot of crap from fans for being a “bad mother” or for being mean to Jon, but I’ve always felt sympathetic towards her. In the books, she’s a little different though. They aged her and all the other characters up on the show, and they made her less severe against Jon. I think she played such a huge role in Robb’s victories in the books that got downgraded on the show because they wanted to focus more on the hot young guy. I was really bothered by that in Season 2 actually because she’s the one with all the POV chapters in the book, not Robb, and yet they gave Robb so much more focus on the show. Like, the whole trading Jaime for her daughters. That was her idea in the books but on the show, it’s Littlefinger who gives her the idea.
Robb Stark - In the books, Robb is like a blank slate. I don’t think he’s really that fleshed out because he never gets an POVs but it worked because none of the “Kings” got POVs. And you really got the sense that he was just a kid who was thrust into this position of leadership that he wasn’t ready for and he needed his mother there to help him. The show really doesn’t capture that. They’ve really tried to make him into the heroic leading man but it shortchanges Cat’s role so much. Then, we get to his “big mistake”, which got butchered on the show imo. What was meant to be a demonstration of how Robb had inherited his father’s strong sense of honor, even if it meant being self-destructive, instead turned into a stupid Romeo and Juliet teen love affair.