The Goldenrod Gym is home to Whitney, who holds the Plain Badge. She uses Normal type pokémon.
The trainers here use Sentret, Meowth, Snubbull and Jigglypuff. Fighting moves are the only supereffective moves against Normal pokémon. If you get a pokémon like Onix or Geodude and teach it Rock Smash, you can use it against these pokémon.
Whitney herself has a Clefairy and a Miltank. Miltank knows a Rock move, so be sure not to use Flying, Fire, Bug, or Ice pokémon against her.
After you defeat her, you will have to talk to her because at first she throws an tantrum and starts crying. But afterward she will give you the Plain Badge, a TM, and the ability to use Strength outside of battle. Strength lets you push round boulders around.
After you defeat Whitney, be sure to get the squirt bottle from the flower shop.