Note on the first book: I've read quite a few reviews from people who intensely disliked the first book, Divergent. Very good arguments were made and I agree with practically everything said: illogical world building, annoying aspects to Tris' character, etc. But, Karen's review puts in words better than I can why I loved it. All bad points aside, it was just plain fun to read and I couldn't put it down. I found myself struggling to stay awake into the wee hours of the morning just to finish it.
Reading Insurgent nearly reached that same level of enjoyment and fun. It picks up right after the events of Divergent. Dauntless is split, factions are at war and Tris is dealing with guilt over her own actions during the battle,killing Will in self defense. Over the course of the book we get to see a little more of the other factions and even the factionless. There are quite a few hints throughout to a big secret kept by Abnegation and coveted by Erudite. So many in fact that the nature of the big twist at the end wasn't really a shock after all. The details were in question Are the factions a social experiment, a vr computer simulation or something else?, but the overall picture The city of factions was not a natural result of events over time, but a specific and isolated construct. was easily deduced.
Did the illogic and annoying characteristics carry over from the first book? Probably, but you'll have to ask someone else about those. Once again I was so swept up in the action and unfolding events, that I was able to pretty much ignore any inconsistency that came along. I'll definitely be reading the third book; I still want to know what happens next.