Although Koga uses Poison-type Pokémon, most of the other Pokémon trainers mainly use Psychic-type Pokémon instead. Just use the same strategy as with Saffron Gym (Dark-type moves like Bite, using Gastly) and you should be able to defeat them easily. For Sandslash, Water or Flying-type attack will work well.
When facing Koga, avoid bringing any Fairy or Grass-type Pokémon. Ground or Psychic-type attacks is best on all his Pokémon, as well as Water-type attacks too on his Nidoqueen. Some examples of Pokémon that are useful against Koga include Butterfree and Drowzee, due to their Psychic-type moves. Gastly can also be used here because of its immunity to Ground and Normal-type moves, as well as its inability to get poisoned.
Koga will award the Soul Badge to you after he is defeated. With the newly obtained HM Surf, you are able to explore more areas that are previously inaccessible. From here, you could either head towards Cinnabar Island straight away, or enter the Power Plant back in Route 10 to catch some Electric-type Pokémon.