The first gym leader is Brock, he gives you the boulder badge and TM34 Bide if you beat him.
He is in Pewter City, and the boulder badge increases pokemons attacks, and lets you use
flash outside of battle. After you beat him the guy guarding the way to Cerulean will move.
If you can beat his Geodude, and get some of Onix away, and have Onix seeded with a
Bulbasaur you should be able to beat him with a pidgey using sand attack, and holding off.
He uses:

lv. 12 Geodude
lv. 14 Onix

The next gym leader is Misty, she gives you the cascade badge, and TM11 Bubblebeam if you beat her. She is in Cerulean City and the cascade badge makes pokemon up to lv 30 obey you, and lets you use cut outside of battle. Basically use Bulbasaur at lv. 20+ again, or a highly evolved Butterfree/Beedrill to beat her. She uses:

lv. 18 Staryu
lv. 21 Starmie

After Misty is Lt. Surge in Vermillion, he uses electic pokemon, so use a Diglett, or Dugtrio from Digletts cave, near the S.S. Anne. He gives you the Thunder Badge, which lets you use Fly outside of battle, and increases your pokemon's speed. He also gives you TM24 Thunderbolt.He uses:

lv. 21 voltorb
lv. 18 Pikachu
lv. 24 Raichu

After Lt. Surge is Erika at Celadon, she uses grass pokemon, so unless you got a charmander, by the secret path(near Celadon, after you come out from Lavender town!), and the station thing to go to Saffron (after Lavender town, under the four trainers!), in that patch of grass you will find a Vulpix level 18-21 if you look hard enough. Train it to level 22 and you should be able to beat them with Ember, and if that doesn't work, use a fire stone. (get one before you beat erika, her physical attacks, and I can't remember, but I think poisonpowder doesn't work). She gives you the Rainbow Badge, which lets you use control lv. 50 pokemon, and increases your pokemon's attack. She also gives you TM21 Mega Drain. She uses:

lv. 29 Victreebell
lv. 24 Tangela
lv. 29 Vileplume

After you head over to Saffron City, you need to beat Sabrina, she uses psychic pokemon, DO NOT use a Kadabra/Alakazam! She will just use Recover! And it isn't very effective, trust me! I would use a Gastly or something!

lv. 38 Kadabra
lv. 37 Mr. Mime
lv. 38 Venomoth
lv. 43 Alakazam

After Misty is Lt. Surge in Vermillion, he uses electic pokemon, so use a Diglett, or Dugtrio from Digletts cave, near the S.S. Anne. He gives you the Thunder Badge, which lets you use Fly outside of battle, and increases your pokemon's speed. He also gives you TM24 Thunderbolt.He uses:

lv. 21 voltorb
lv. 18 Pikachu
lv. 24 Raichu

After Misty is Lt. Surge in Vermillion, he uses electic pokemon, so use a Diglett, or Dugtrio from Digletts cave, near the S.S. Anne. He gives you the Thunder Badge, which lets you use Fly outside of battle, and increases your pokemon's speed. He also gives you TM24 Thunderbolt.He uses:

lv. 21 voltorb
lv. 18 Pikachu
lv. 24 Raichu

After Misty is Lt. Surge in Vermillion, he uses electic pokemon, so use a Diglett, or Dugtrio from Digletts cave, near the S.S. Anne. He gives you the Thunder Badge, which lets you use Fly outside of battle, and increases your pokemon's speed. He also gives you TM24 Thunderbolt.He uses:

lv. 21 voltorb
lv. 18 Pikachu
lv. 24 Raichu