"Pokemon LeafGreen" is one of the remakes of the first generation of Pokemon -- the series that started it all. It is available for the Nintendo Gameboy Advance. In both the remake and the original, you can find Pokemon to battle all over the region, in tall grasses and in the water. Pokemon such as the Water-type Staryu are only available to capture if you have the Super Rod, the best fishing rod in the game.
Go to Lavender Town and travel south past the Gatehouse. Enter the house just south of where the Snorlax was and talk to the fisherman. Talk to him, say that you love fishing, and you will receive the Super Rod.
Go to Pallet Town, Cinnabar Island, or Vermilion City.
Walk to the edge of the water. Press Start to bring up the menu, then scroll to the Key Items and select the Super Rod with A to use it. You may not get a Staryu immediately; if you do not get one the first time, just beat the Pokemon you do get and keep trying.
Weaken the Staryu when you find it with your own Pokemon by hitting it with low-power attacks until its HP drops into the red. Use a Great or Ultra Ball on the Staryu to capture it.
If you have not battled the Snorlax to the south of Lavender Town, you will need to beat him first before you can get the Super Rod. Get the Poke Flute from Mr Fuji in the tower in Lavender Town to wake him up for a battle. Staryu is only available in the LeafGreen version; it does not appear in FireRed.