This story has been made the shortest as possible but i might have missed some important parts so soz for that. These pics were most likely taken from either
http://www.animeinfo.net/laputa or
http://looney.physics.sunysb.edu/~daffy/laputa.html 
hehe i found these pics and links on my old backup discs so im not quite sure if its accurate...

Well the story starts off with Sheeta on an airship where all of a sudden air pirates barge into the ship and full trash the place. Sheeta tries to run away for some reason but slips and falls down onto the earth. 

But after a while (while shes still falling) her pendant releases some sort of power and she starts floating downwards.

Pazu, who was working in the mine (thing) at that time saw her falling down and decided to go see what it was. He then saved her.

Then you have this long bit of nothingness where the two learn about each other - basically how theyre orphans, how pazu's dad found laputa but no one believed him and about how the pendant sheeta has is an heirloom and how its a very special stone that can only be found/made in laputa

They run away from the air pirate dudes and government dudes (the same dudes who were in the airship with sheeta before the air pirates came in) for quite a while during that bit of nothingness

Finally these government dudes try and take her away. Pazu tried to save her (what a cutie) but couldnt

Sheeta is threatened to stay with the gov dudes and help them find laputa and pazu is sent home

Sheeta finds out (i dont remember when exactly she finds out) that Mooska (one of the government dudes) is from laputa

When pazu gets home he finds that the airpirates are there and theyre trying to eavesdrop on the gov's plans and stuff and decide to head off to sabotage their plans. Pazu begs to go to save sheeta and they let him.

While that is happening, sheeta reminisces and accidentally recites this sos spell that can only be used with the pendant that her grandmother taught her

So a robot comes out to save her but coz its so scary sheeta tries to run away and the gov ppls are trying to kill the robot and stop sheeta from running away coz she has the pendant that is very valuable in finding laputa (coz it points to its direction). 

Sheeta runs up to the very top of the building where Pazu happened to fly past and saved her. Sheeta drops her pendant and Mooska gets it

Sheeta and Pazu join the air pirates to find laputa. Pazu realy wants to find it to prove that his dad was right. Sheeta wanted to find out of her past or.. something

And heres another long bit of nothingness where the gov ppls and the air pirates are racing to find laputa.

Theyre almost there so Pazu and Sheeta set off in another little airplane

They get to laputa but the air pirates are prisoners of the gov and the gov ppls are really concentrating on getting all the jewels of the place

Mooska manages to get Sheeta but not Pazu

Pazu sets the air pirates free and they give him weapons to help get sheeta free

Mooska uses sheeta's pendant and his laputa language decoding book to become king of laputa. He kills lots of ppls in the process

Then you have this bit where sheeta is running away from mooska and pazu is trying to find sheeta

Mooska threatens to kill sheeta

Pazu finds sheeta just before it

Mooska lets them talk for one minute or.. forgot how many minutes

Pazu and sheeta decide to recite the destruction spell

The buildings of laputa are destroyed but the natural structures stay

Pazu and sheeta leave the place safely and find that the air pirates are still alive with jewels.

They head off somewhere (not with the air pirates) and thats the end of the story

The moral of the story i think is that natures better than technology.. well in the canto version sheeta recites a poem that kinda means that near the end - right before pazu comes along

.