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Gold and Silver: Shiny Pokemon...the mystery
First of all, don't get your hopes up on turning on your game, and then after an hou=r or so getting a Shining Pokemon. In fact, the chance of you finding a Shiny Pokemon are 1 in 655361! That is a pretty small chance of encountering one!

Of course, don't get your hopes to down, because you can find a shining Pokemon in the wild no matter what! In the Lake of Rage, there is a Shining Gyrados sitting in the middle of the lake. All you have to do is go up to him and press A to start the battle (I advise you save prior to battling)

And probably the easiest way to get a Shining Pokemon is to get the Shining Gyrados and then breed him/her with another Pokemon of the opposite sex, because the genes will transfer over to the new Pokemon.

Also, if you can manage to get a Shining Ditto, then you can make almost any Pokemon Shining. To get a Ditto you first need to have a Red, Blue, or a Yellow version. Start out by getting yourself a shiny Pokemon, for example the Red Gyarados up at the Lake of Rage. Now delete all its moves except for one that doesn't do damage like Leer or Rest that appears in Red/Blue/Yellow. Now trade the shiny Pokemon back to Red/Blue/Yellow, but don't worry if it looks normal. Now teach it TM31 Mimic and buy as many Ultra Balls as you can (or the Master Ball if you did not use it), and find a wild Ditto. When you run into one, use Mimic to learn Transform. When Ditto uses Transform, it will either know Mimic or Transform. Either way, it will eventually use Transform again. Be sure not to get its HP down and throw an Ultra or Master Ball at it to catch it. The special thing about your new Ditto is that it will have permanently copied the shiny Pokemon's genes. Go to a Pokemon Center and heal it right away because its HP will be a little off and sometimes it will have more HP than it can possibly have. Now trade your Ditto back to Gold/Silver/Crystal and you will have a shiny Ditto. Now breed it to have a better chance of getting a shiny Pokemon than before.