The story continued in chapter 2. Chapter 3 as well, only with a completely changed cast. Whether you need to watch it it;s up to you, I think you can consider chapter 1 as a complete story, maybe with chapter2. I haven't seen chapter 3 myself but I do not feel I am missing much, but thats just my pov. In terms of story, I probably am missing more drama, but I don't feel particularly interested in watching season3 with a different cast, since you really grow to like the vast of the first two seasons.
There is now also a remake of the drama, done by the same writer. It is available on viki and is called New My Fair Princess. It has 90+ episodes as it incirporates all three seasons but this new version is vastly different from the original one.
