Sadly, there is no way to get them all three. Except by you playing the game 3 times and trading with another game like Leaf Green, Fire Red, Sapphire, Ruby and Emerald.
It is an awful lot of work.
But lets start with the basics, you get one (1) legendary beast without needing to re-play the game from scratch and deleting your progress, depending on your starter.
If you chose Charmander as a starter you will get Suicune at the end of the game.
If you chose Squirtle as a starter you will get Raikou at the end of the game.
If you chose Bulbasaur as a starter you will get Entei at the end of the game.
Key is, you get the legendary beast that your starter is weak against. I suggest you SAVE the Master Ball for this pokemon and this pokemon ONLY!
Why? Because he will run from you as soon as you encounter it.
If you do not have the masterball because you used it on Moltres, Zapdos or Articuno, then make sure when you encounter him the first time (usually on route 23 is the first place you will see it) look into your Pokedex to see where he is located.
SAVE YOUR GAME OFTEN!!!!! If you happen to knock him out he will be gone for good!
Carry LOTS of Ultra Balls and Great Balls with you, and Poke Balls just in case. Make sure all of them amount to at least 60 pokeballs. DO NOT burn or poison it, the damage you inflict on it will remain next time you encounter it in the wild!
There are tricks to do to make its capture as easy as possible, here are a couple of YouTube videos that help with that.