Before you go to the Ecruteak Gym, you have to go to the Burned Tower in the northwest of town. Here, you will get your first glimpse of the legendary beasts: Raikou, Entei, and Suicune. You will also have an encounter with Silver! After you fight him, go and talk to Morty, and then head downstairs. When you approach the legendary beasts, they will flee.
Once they do, they will be roaming around Johto. If you see its icon on the map and go into tall grass or another area with random encounters in the place where its icon is, you might encounter it for a battle. If you encounter one, you can run away and it will still be roaming. If you knock one out, then you won't be able to find it again until after you defeat the Elite Four. If you reduce its HP, it will still be reduced until the next time you encounter it. You should try to capture them!
Suicune is more elusive, however, not roaming Johto like Raikou and Entei. Instead, you will encounter it in various places later on, and will be able to catch it later on.
But anyway, once the legendary beasts are roaming, you can go to Morty's Gym. He uses Ghost type pokémon, so don't bring Psychic or Ghost types. Use Dark or Ghost moves against them.
Once you have the Fog Badge, you can use Surf outside of battle. This opens up quite a few new regions to you, so be sure to explore.