I read this book for the first time when I was about 9 or 10. Back then I liked the story and the fact that the main character was a girl. Pretty much everything even mildly inappropriate slipped right over my head, and now that I'm rereading it at 13 I'm actually picking up on the stuff/understanding it.
You should know that it does get violent, especially near the end. There is death, and two of the main antagonists are Lyra's parents. Iorek, the bear, was kept in the town where they found him because the townspeople were keeping him drunk, so there's some mention of that. They talk a little bit about an affair, and they briefly discuss castration. Like I said, most of that stuff went right over my head when I was younger, so it should be pretty safe as far as that goes. And if you're afraid that your kids will be affected by the religious stuff, don't worry about it; when I first read it, I barely realized the religious aspects of the book. Something the later books deal with is the concept of death and what happens afterwards, so that could prompt some "What happens when people die?" questions.
Lyra is a strong female lead, and she's adamant about doing what she thinks should be done. She can be stubborn at times, but that really just makes her more believable.
With all that being said, I think this book is definitely a great read. If your kid is younger (10-11) and wants to read it, I'd say go ahead and let them; if they want to, they can read it again later, and pick up on what they didn't. If they're older, like 12-13 or above, it's definitely appropriate.