Items needed:

Poke balls (stock up with a lot and different kinds for legendarys.)

When you are engaged in a battle with the Pokemon you want (even legendary or one of the three Regis) weaken it until it's HP bar is red, throw a poke ball at it. If you knock it out, reset the game and experiment with the moves to see what is effective and what is not and then when you have it with low HP throw it. (If it is legendary or rare, use an ultra ball). then as soon as the ball opens, hold A and B button. If it doesn't work, keep trying, as it will have a higher chance of catching it.

This works for me, so hopefully it will work for you ;)