This is the first book in a series, it's quite engaging, and if you child likes it, they will want to read all three. However, you'll want to read it, too, as there is quite a bit that you'll want to talk to you child about.
The series contains a lot of violence--from fighting, wars, and torture (of animals, and in later books, of people). This is handled mostly pretty well, but it bears pointing out (spoiler alert) that the protagonist is a hired thug to rough-up or kill others (although she has a conscience about it). There are mentions of rapes, visions of torture and murder, etc. in the series (particularly the last one, "Bitterblue")--all in all, pretty grim.
The part that bothered me in general is that in all the books in the trilogy, there is the general message that sex is a casual thing between friends (particularly prevalent in "Graceling", the second book, where two characters get together just to "hook-up"). One can admire that there are many references to taking an herb to avoid pregnancy (some notion of "safe sex"), but as this is in a fantasy land, there are apparently no sexually-transmitted diseases, so no concern over that. In general, the not-so-subtle take-home message from the books in the series is that sex feels good and is fine between friends so long as you do something not to get pregnant, which isn't something I'd want my kids reading without talking to them about it.
Unlike some reviewers, I don't subscribe to the notion that just because kids today are barraged with sex and violence in the media, that it's appropriate for them. I would not recommend this series to those under 13. (If they were made into movies, they would definitely get at least a PG-13, based on violence and sex.)