Are the Movie and the book fairly similar?

Definitely not. Some of the major plot points are the same but there were many other things that were changed, and characterization between book and film is completely different.

John and Mary had never met. They were like two hummingbirds who had also never met.

Pretty much no. Characterisation (mostly strikingly Fanny's characterisation is completely turned on its head), style, some of the themes and even some of the plot is changed (the slave trade in the Caribbean is hugely amplified from the one sentence in the book, Jane Austen seems to make a point of not writing outside of a small community of ideas and issues) Nothing against the movie, which is enjoyable in its own rights. However, there is a chance that if you enjoyed this movie, you may find the book dull.

Others have already answered this question, but I'll put in my two cents.

No, the movie is not much like the book. One major difference is that Fanny is a completely different character. In the book she is shy, timid, introverted, and has a rather delicate constitution (she was a first quite afraid of horseback riding, and only rides one for the sake of her health and never at a gallop), while in the movie, she is robust (fearlessly riding her horse at night in the rain) and speaks her mind. In fact, her actions in one scene contradict what is said in another scene.

I suppose the plot is (sort of) the same...