How to catch the elusive Raikou, Entei and Suicune:
These three Pokémon (or two if you play Crystal) were the first of their kind in that they roam around the game and flee battles quickly. The only way to encounter them is by getting lucky and bumping into them in the wild, and they can easily Run or use Roar to escape. Catching them is no easy task, but it’s doable even without a Masterball.
Step One: Unleash Them
Go to the Burnt Tower, approach them and let them run away. Don't panic - they're supposed to do that!
Step Two: Build Your Team
These Legendaries are very tricky because they can Run away whenever they want, or use Roar, and if you accidentally kill them, they won't come back. That is why the following attacks are needed:
– Catch a Scyther/Scizor from the Bug Catching Contest, or Farfetch'd on Route 38/39.
Mean Look/Spider Web
– The Gastly line and Spinarak line know these respectively. Gastly can be found in the Spout Tower at night and Spinarak on Route 30 and 31 at night (but only in Silver).
Hypnosis/Spore/Sleep Powder/Sing, and Fly
– For these, check the Serebii Attackdex
A Gengar with Mean Look and Hypnosis and a Scyther/Scizor with False Swipe should be perfect. The Pokémon with Fly is only there for transportation, so it can be anything. Alternatively, put all these moves on a Smeargle if you can manage it. Raise your team to at least level 45. You want them to be faster than the beasts.
Step Three: Stock Up
Buy a lot
of Ultra Balls and get Fast Balls from Kurt. Have at least 15-20 on you at all times. You might also want to buy some Repels so you won't have to waste as much time battling wild Geodude once you know where the beast is.
Step Four: Find the Legendaries
This will first depend on chance, as there is no way to track them until you have encountered them randomly in the wild. Don't worry if they get away the first time.
Step Five: Chase The Legendaries
Once you have seen a beast at least once, you will be able to find its current location through your Pokédex. However, once you enter a city, route, cave, building, or simply turn your game off/on, its location will change. So it won't stay in the same place for very long, and catching up to it is very hard.
To begin the hunt, keep your Gengar first in your party, check the beast’s location and Fly somewhere nearby. There is a slim chance that it will change places to be nearer you or even in the same place, because they cannot jump from one end of Johto to another. Rinse and repeat until you find them; it's a long game of cat and mouse, but you'll also learn how to anticipate their movements.
Step Six: BATTLE!
Once you encounter Raikou, Entei or Suicune, start off with Mean Look or Hypnosis. Either move can be good or bad; Mean Look prevents it from Running, but it can still use Roar. However, while Hypnosis prevents it from doing either, Hypnosis has low accuracy, and the beast may wake up immediately and Run or Roar.
Either way, ensure the Legendary is put to sleep, then bring in Scyther. Use False Swipe as many times as possible; this attack will never kill a Pokémon, so it's always safe. If the beast wakes up, switch back to Gengar if possible to put it back to sleep before it gets away. When False Swipe doesn't seem to be doing any more damage, it means that the Legendary only has one HP left, and that's your cue to start throwing Ultra Balls and Fast Balls! Keep Gengar out on the field during this time in case it wakes back up. It may take a good few battles with each beast to get them to this point.
If they escape, don't give up, because all's not lost. Any damage you have done to it in the past is still there, so if you know it only has one HP left you don't really need Scyther anymore. Just keep using Mean Look/Hypnosis/Ultra Ball, and you should catch it eventually.