My favorite area is the Egg Corridor, because of many things.
Chiefly, it was the first area I ever used a level 3 polar, and it was the first area where I could really kick some butt.
Secondly, it got the game's momentum kicked off and going. After all that dialog and those cut-scenes, it was nice to be able to sit back, relax, and enjoy the Cave Story platforming goodness.
Thirdly, I liked how the computers in the room with the rocket launcher gave you clues to how to get through the forcefield. I felt like I could connect one thing to another, and that it felt like I was doing a quest in an RPG, without any help from a journal or a marker.
I also liked the enemies. The flies, the elephant things (as impossibly easy to ass by as they were), and the classic fluffy jumpy things, made for a delightful Cave Story environment.
Additionally, there was a good degree of difficulty befitting a first level. To go through it normally took no extra skill, but to get the life capsules required a bit of finesse. Also, considering I'd recently played a game that was all about jumping, it was a little tricky for me to get used to the way Quote jumped, so navigating through the corridor was a nice little challenge.
Lastly, I liked the music. It is one of my favorite tracks in the soundtrack, and it gave me a feel of wondrous adventure, and an opening up to a new world. It set the mood very well.
So that's basically why I like the original Egg Corridor area.