Well, to get the whole feel of Kingdom Hearts, it's best to play all the games, as they will complete the huge messy puzzle in our head called "STORYLINE"
I will try to answer without spoilers. Maaaaybe just a thing or two to make you curious xD
The events in Kingdom Hearts 2 happen aprox. 2 years (perhaps a bit less) after KH1. In this time interval, the events of Chain of Memories and 358/2 Days occur.
When you start KH2, you'll get confused about "who the hell am I playing as??" "why am I here?" "why was Sora in that place?" "why is everybody calling him R O X A S?" "dafuq happened with Riku??" "who's that smexy redhead guy?" "who's that blonde chick?" "why is everybody so angry/sad?" "why is Sora crying?" (my experience) and many other things. Of course, they will keep you curious enough to finish the game, but the game doesn't really come with explanations.
Now, 2 years is quite a lot of time, right? Sora was left in some dark world in the end of KH1, finds his way to a certain spooky castle... But what about Riku? He and King Mickey were trapped in the Realm of Darkness, right? Hmm... perhaps something happened to them too. That's what Chain of Memories is about. Story-wise, it's very important. Many new characters are introduced and the KH universe becomes larger. From my experience playing CoM for the GBA, I loved it, even the rather challenging gameplay. Remember, real gamers don't complain, they master all play styles ;P
Also, after KH1, I was such a Riku hater, man, talk about "hate" with a capital "H"... And then CoM changed everything for me. EVERYTHING.
Be warned, a lot of people die in CoM. But there's a happy end... well, kind of.
358/2 Days is the sad story of a person that was never meant to exist. No wait, of a BUNCH of people who never were supposed to exist. Yet they still are... As a game, it really made me feel his daily experiences, the lies he was always told, the false beliefs he and the other members had. I recommend the game and not the movie for that amazing, saddening and unexpected experience only a player can have. Even so, to deepen the story and also explain lots of the KH2 events, watch the movie, it's yet another side, another story. Oh, and that secret ending shows something that happens at the very end of 358/2 Days. Something really cool, yet also sad. This is a sad game. A good, sad game.
Be warned, a lot of people suffer or are left suffering after Days.
You will recognize characters from CoM and Days when playing KH2 and you will understand their sometimes desperate actions. Until KH3 comes out, which marks the end of this saga, you have plenty of time to cover all games, so why not? And you want to clear up your mind, I bet you're super-confused about everything and KH2 will make it only worse xD