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Your rival is named Blue, Professor Oak is his grandfather.
Team Rocket is the gang in this game, their boss is the final gym leader.
Mewtwo is the evil legendary. It's a man made Pokemon like Genesect. You have to beat the game first and then go into a blocked off cave in Cerulean City to find it.
Some Pokemon, like Kangaskhan, can only be found in the Safari Zone. You don't catch Pokemon by battling here, you have to use bait and food to lure them first.
The fourth gym is grass type. The leader is a girl named Erika and in this city you can spend in game currency to gamble for prizes, like rare Pokemon.
If you surf along the edge of one of the late game cities your game fucks up. Hopefully this gets fixed in a day 1 patch.