(FYI This will be less about things I noticed and more just me gushing about how I felt about her character execution... sorry lol)
Initially I didn't care for Madoka's character much, mainly because I was expecting her to become a magical girl much, MUCH earlier and I'm not normally a fan of characters that spend most the time crying, though by the ending of S1 and after my 2nd viewing I came to view her quite a bit differently. Funny enough, my initial impression just made me like her even more when I finally came to understand what was going on and just how well her role in the story was executed. As the main character of the most deceiving show I have ever seen, it makes sense that my first impression would be almost entirely wrong.
Madoka was like a sponge for all the despair in the series, everything that happened to the people around her she felt. She doesn't cry as much as she does because she is weak willed, she cries because she is genuinely a caring person who can't turn her back on others suffering even when she herself does not know how to make it better. And as a result of her facing that and letting herself feel what happens to others rather then bury it she is able to become (imo) the most mature character in the entire series. That role of being an observer fits rather well when you realize that most of the series is just building up to her final decision so that you can understand the weight behind her wish and decision (I really appreciate the focus the story has on why the decisions are being made rather then just going from one plot point to another) I'm also am happy that TRS took it one step further and explored whether it was right for her to have to have made a decision like that rather then just romanticizing her sacrifice.
As someone who loves character driven stories I rather like that the focus ended up being on Madoka's relationship with the girls and their relationships with each other rather then about them just overcoming and fighting witches. That added a lot of substance to this series for me and makes me respect Madoka as a character quite a bit.
Her relationship with Homura interests me to no end. The duality that exists between the two is simply amazing. It's funny how people spend so much time arguing whether or not Homura and Madoka are in love, since the real scary question comes once you accept they are in love and start questioning what the nature of love even is, which is what I believe the story intended (basically a deconstruction of the concept). The people who worry so much on whether or not it should be classified as yuri are just going to keep getting trolled by Gen if he ever makes a sequel. Animus, you may be frustrated by the fans, but Gens laughing at them. He is intentionally tormenting people and you either have to applaud how well he does it or curse what a bastard he is for doing it. For me I do a bit of both.
It's pretty much like Gen's using people's love of yuri to bait them into a trap and make them actually think. That sick magnificent bastard 
My favorite Madoka moment is during the third timeline 3 in episode 10 when her and Homura are talking after the Wulpurgis fight, and Madoka asks Homura to prevent her to go back in time to stop her from becoming a magical girl, and also to prevent her from becoming a witch before she does that. Don't get me wrong it was completely soul crushing, but god damn was it powerful Madoka expressing happiness at Homura finally calling her by name right as her soul gem explodes was...
and that she used Sayaka's grief seed to do so just made it all the more impactful.