For me the answer changes from books to TV shows. In fact I think for me most of the answers are different for me based on the book or on the TV show. I very much prefer the story the way the books lay it out, even though at times it is super hard to read. The books are so dense especially books four and five as I sit reading each chapter twice, with a GOT app open on my phone and the The Wiki of Ice and Fire on the laptop. This is fun, but a lot of the time it feels like work lol.
So my Favorite Characters from the books first, and then the TV show:
Without a doubt Daenerys Taragaryen is the most interesting person in the books. I think that is true because she is the one character we see grow. We could make that same argument for Arya Stark, but for the most part the books are full of people who are what they are. When we first meet Dany she is a scared little insecure girl being taken advantage of by her crap hole brother. By the end of book five she has grown a lot and of all the characters in the books her change is largely a positive one. For me Dany is the one person, save Sansa Stark, who tries to do the right thing. I like her because she is doing the right thing not for herself but for how she sees the world. That may make her naive, but it is a very endearing quality. I think from book one to book five she is the character that changes the most outside of Jon Snow, but her transformation is far more drastic.
I love Tyrion in the books too, and he is smarter than everyone else in Westeros by at least three times. One of the only Hand of the Kings to not get murdered, and without a doubt the best Hand we have seen on TV. His role wasn't even cast, George Martin said its Peter Dinklage of no one. He is by far the most entertaining character on the show, and the scenes in season two with him and Cersei are among my fave scenes of the whole show. Tyrion is a great character and as much as he is out for himself, he is also trying to do the right thing. Even Tywin sent him to King's Landing to get Joffrey and Cersei in line. He gets screed over a lot and is the ultimate under dog, despite his standing as a member of House Lannister, but I think that is why a lot of GOT super nerds (like myself) like him so much.