Ok, I'm going to do this from memory, so if I miss anything, tell me.
So first, beat the elite 4 and the champion. After that, you must have 60 different kinds of Pokemon completely registered in your pokedex. When you have the pokes registered, go to professor oak. He will tell you to go to one island. Go to the port in vermillion city and take the ship to the island. Talk to Celio. He will tell will tell you that he needs a ruby. Go to Mt. Ember and talk to the grunts. After you battle and beat them, go inside the cave. After you work your way around the cave, you should eventually find the Ruby. After you return it to Celio, he will tell you that he needs something else, the Sapphire. Then he will give you the rainbow pass. That allows you to go to the other islands. Take the ship to go to four island. And complete the icefall cave mission with Lorelei. Then go to five island and battle the team rocket grunts. One of them will tell you one of the passwords you need to get into the rocket warehouse. You then go to six island and go into the dotted hole. To get it you need to use cut on the door. I will provide a link that shows you how to get through the hole. After you get through the hole, you will see the sapphire. A scientist will take the sapphire and will tell you the other password to the rocket warehouse on five island. After you get through the rocket warehouse defeating all of the grunts, you then battle the scientist. After you beat him, he will give you the sapphire. When you give the sapphire to Celio, he will put it in his pc. This will allow you to trade Pokemon between Pokemon Ruby and Sapphire. You then would have completed the mission of the seven islands, even though you only went to six :P.
Hope I helped!