Overall, I like the new episodes very much. It feels a little odd that the stories are so short; I can see places where in old Dr. Who, there would be secondary plotlines, but here they are only brief scenes, as when, in the second episode, Rose has a conversation with the blue plumber and learns they don't speak unless given permission. It was like shorthand for themes explored in the old series.
The first episode felt familiar all the way through: beginning with the companion's everyday life before she meets the Doctor, and then seeing her world turned upside-down by alien invasion. She was in the basement of the department store and I immediately thought, "Ooh, Autons!"
I got the impression the Doctor had only recently regenerated--there's a moment when he sees himself in a mirror at Rose's apartment and makes some comment about his ears. This idea wasn't contradicted by the pictures website-guy Clive had of this Doctor in various times, because those visits to other times could be in this Doctor's future.
Rose is a very traditional sort of companion--spunky, questioning, willing to go along with the Doctor's fey moods. I liked her a lot. She's another way in which the new series felt comfortable and familiar; she's the one who speaks to the "ordinary" people while the Doctor deals with the Big Picture, and sometimes the "ordinary" people turn out to be the key.
One thing that bugged me a little--in all three of the first episodes, the Doctor doesn't do the saving, someone else has to save him. It's true this gives the viewer a way into the story through the "ordinary" characters, and makes them active as well. I'm hoping, though, this doesn't become the sole formula; I like the idea that even the Doctor needs help, but at the same time he has to be the sort of character who can always pull a rabbit out of a hat, that's what he is. In "Rose," Rose saves him from the Autons, and coincidentally kills the Nestene Consciousness. In the second episode, the tree lady sacrifices herself to aid the Doctor in saving everyone else. In the third episode, Dickens thinks of a way to defeat the gaseous creatures, and Gwyneth sacrifices herself to destroy them completely. This sort of thing is by no means unknown in the original series, but it seemed repetitive three times in a row, especially seeing all three episodes in a single night. And it implies one must die to be heroic.
The Doctor does put the disparate parts together and solves problems that way, but he seems less physically active than he might be, slightly less prone to always have an answer. Not necessarily the best answer, but an answer. I'll be interested to see how this develops.