They are connected, but not in a way that you lose much if you don't play them sequentially. Some of the characters are related to each other, and some of the lands you visit will be mentioned in other games.
The first King's Quest game I played was the sixth one, and I think it is widely regarded as the best. I'm not sure how well it holds up now, but to a 10-year-old with an entire summer vacation to play it with a friend, it was really magical. It draws on a lot of fantasy tropes and has some fairly epic puzzles, so if you let your imagination run wild I think you will get fairly immersed in it. That might be the 10-year-old in me talking, though.
I picked up the entire KQ collection either through a Humble Bundle or on steam/GOG for pennies a year or so ago, so cost will be a very small barrier to entry.
oh, and KQ VI has this:
https://www.youtube.com/watch?v=69M2iiI__XQ