Before entering the Gym, make sure you have a Pokémon that’s at least level 15. This is a requirement to enter the Gym, but thankfully this is a pretty easy requirement for you to meet and you should have no trouble with it.
You can actually take on the Gym before going to Route 25, but it’s recommended you get the extra EXP. It’s also strongly recommended that you teach your partner Pikachu Zippy Zap or your partner Eevee Buzzy Buzz from the Move Tutor in the Pokémon Center before entering the Gym. It will make your time here much easier, although arguably too easy if you’re looking for a challenge.
Beyond that, having Grass-type Pokémon such as Ivysaur, Bellsprout, or Oddish is a good idea, although be wary of a certain Psychic-type move that can do some decent damage due to Poison being weak to it.
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You have to at least fight the first Trainer, but the other two Trainers can be avoided by taking the long path, similar to the Pewter Gym. They use a variety of Water-type Pokémon, as to be expected.
After you’ve made your way to the end, you’ll have a chance to battle against the Cerulean City Gym Leader — Misty.
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If you were expecting Misty to lead with Staryu, surprise! She instead has a Psyduck to cause you headaches instead. This thing packs both Water Gun and Confusion. Neither should hit overly hard, although Confusion will deter any of your available Grass-type Pokémon due to all of them also being part-Poison-type, which is weak to Psychic.
Her Starmie is a much greater threat. Despite being part-Psychic-type, it doesn’t have any Psychic-type move that is affected by weakness — it only has the level-based damage move of Psywave — so your Grass-type Pokémon are a solid counter. It still hits hard with Swift thanks to its high Special Attack, and any Pokémon that isn’t resistant to Water is at risk of getting hit hard by Starmie’s Scald. Scald is a very strong Water-type move that also has a chance of burning you, so it is nothing to mess with. Your Pikachu or Eevee with its Electric-type move should still do just fine against it, but if you are going in without those and without a Grass-type, you’re going to have a rough time.
After beating Misty, she’ll reward you with the Cascade Badge, and on top of that, she’ll give you TM29 Scald — a strong, 80-power Special-based Water-type move that has a chance of burning the foe. It’s a great move to teach any Water-type Pokémon you have, such as Psyduck.
Anyway, now that you’ve beaten the Gym, you’ve got a little bit more to do in Cerulean City.