You'll obtain your starter pokemon in the home town lab in New Bark Town (Johto) or Pallet Town (Kanto), after you complete the starter's island quest.
To do the quest, speak to NPCs around the island and do what they ask you to; once you obtain pokeballs/potions/pokemoney/a pokedex, you may depart to Kanto or Johto, where you will be able to select your starter pokemon.
To further obtain more pokemon, battle with one in the grass, weaken it (but don't completely defeat it), then throw a pokeball at it, to catch it. When your reputation is 30, you can trade with other players to obtain more pokemon. Also, you can buy pokemon available on the pokemart (no rep required).