How and where do you catch mew and the legendary dogs?

Depending on Which Starter Pokemon you chose, you'll get either Entei, Suicune or Rakou. If you chose;

Charmander, you'll get Suicune
Bulbasuar, you'll get Entei
Squirtle, you'll get Rakou

The legendary Dogs will roam throughout Kanto, just like Latios or Latias in Ruby and Sapphire.

You cannot obtain Mew as of yet without cheating. However, the new game, Pokemon Emerald will have a way for you to obtain Mew.

You can only get Mew by an action replay. It dependes on wich Starter Pokemon you pick. If you pick Bulbasaur then you will find entai, if you choose Charmander you will find Suicune, AND FINALLY if you pick squirtile you will find Raiko.