Kratos and Ashley. Just kidding, for me it's Agro.
It's hard to say since I'm not really sure how I'd define "care about". I mean, first and foremost would be an self-created character I've made in an RPG such as The Elder Scrolls or Warcraft. I'm one of those people who spends a lot of time in the character creator so I become really invested in my own creations. Even "Cinnamon" Shepard would count, but I think the spirit of the thread is about developer created and controlled characters. The earliest one I can recall that I didn't just think of as cool is Mission Vao, the blue teenage twi'lek from Knights of the Old Republic. I thought she was funny and spunky, but with a troubled past that added to her character. She followed the scoundrel archetype and not unlike that Solo guy, she was distrusting and a bit of a cynic, however she was also loyal to her friends. I think she was looking for something or someone to believe in, and so when my Revan came along, she provided some hope for the young girl. It was fun to see her grow from the earlier part of the game through to the end. And it the dark side near-ending is still something that haunts me. That's was the first time I can recall a game affecting my feelings.
either garret from thief, solid snake, or kyle katarn. i also cared for the shepard i created also a few other characters from mass effect like tali, thane, and legion (who along with thane are complete badasses) which is why i was pissed sheperd died at the end of mass effect 3 but they weren't the first characters i cared about..
I'd have to say Shaun and Ethan Mars from Heavy Rain. While the game took awhile to ramp up for me (I had to explore EVERYTHING), I found myself legitametly rooting for the father and hoping he would ensure the safety of his son. One of only a handful of games that made me root for them and jump for joy when I succeeded.