I have watched a little of the anime and read all of the manga. Personally, the latter is my favourite as the anime does not delve deeply enough into the characters in my opinion, though is still an enjoyable romantic comedy. The manga is probably one of the best ever drawn- Takaya's characters are complex, but not so much that it is confusing. Everything is done gradually, the only exceptions are when we need immediate information- like the start of the series or like sudden revelations.
Everybody is believable and human eventually, and while the story starts out as a simple, lighthearted comedy drama which has just enough seriousness to remain interesting, the art evolves beautifully as time goes on and so does the story- we learn more about the characters. We laugh for them and cry for them. We understand and empathise with them. Tohru begins to engage more actively with the plot, turning the world of the Sohma clan upside-down. Had the story remained at the same level of depth, I would have lost interest, but this story grows with the reader, and has a positive but still realistic message behind it- it's too late to change the past, but you can still make the future bright. Of course, this doesn't mean that there are no more comedic or heartwarming moments after Tohru's decision, but they are better spaced out, and the tone of the story is more mature- in a good way.
Overall, I would say that this is a masterpiece that everybody should give a go. Don't be put off by the cliched premise- the story is so much more than 'oh dear, I turned into a fluffy animal, let's fall in love'. Takaya has a deft use of foreshadowing, and the art is fantastic- the characters are still wonderfully varied, though people unfamiliar with the cast may get confused. It's worth many, many re-reads.