For time's sake, I'll skip the first two badges because I used my Starter until then.
-Went to Vermilion City, traded Spearow for my Farfetch'd.
-Grinded Farfetch'd to go from Vermilion from Cerulean, through the Rock Tunnel.
I'll explain my strategy from here: From Lavender to Celadon, bought a Poke Doll, went back to Pokemon Tower to get my Poke Flute, woke Snorlax, went down Cycling Road to Fuchsia City to get Surf, Strength, and most importantly a Good Rod.
Good Rod was needed to catch my Goldeen (evolves to Seaking) and Poliwag (evolves to poliwhirl, which I needed to trade for Jynx). I got both and collected 3/4 member of my team.
Finally, I went back, super overleveled to fight LT Surge.
SURGE (Badge #3)
Farfetch'd learned Slash right before Surge, so i was able to get guaranteed critical hits in order to defeat Surge. Farfetch'd Speed makes Slash crit 100% of the time. So yeah, Slash for the win.
ERIKA (Badge #4)
Again, very overleveled, so I went straight for Erika with Farfetch'd. As Peck was too weak, I taught it Fly. Erika is never hard when you have Flying types.
Goldeen--->Seaking!
KOGA (Badge #5)
I went straight back to Koga so I could be able to surf to Cinnabar and get my Weezing. Farfetch'd Slashed Koga for another win. I do use my other Pokemon, but Farfetch'd wa most important. Seaking was too.
-Caught Koffing and grinded it to Weezing! My whole team has been collected!
BLAINE (Badge #6)
With a Water type on my team (Seaking!) I could finish Blaine up easily. Seaking Surfed all over him for an easy win.
Now it's time to fight Sabrina.
SABRINA (Badge #7)
With a resistance to Psychic attacks, Jynx was a good choice for this Gym. I had Jynx use Ice Beam to finish off her Pokemon.
GIOVANNI (Badge #8)
I smiled when i saw that I could use Seaking for this run. Why? Horn Drill. For Gio, I gave Seaking an X Accuracy and killed everything with Surf and Horn Drill.
-Smacked my Rival with Seaking only. Gen 1 OHKO moves are so overpowered!!!
LORELEI
One X Acc, one X Speed. Game over, Lorelei!
BRUNO
Same strategy here, except Seaking surfed the Onixs to save PP.
AGATHA
I pulled out Jynx for this one. I needed an X Speed for Jynx, then it swept Agatha's whole team with Psychic.
LANCE
Seaking solo'd this fight. Horn drill for Gyarados through Aerodactyl, then Ice Beam for everything else.
CHAMPION RIVAL
Oh boy, we're almost done! Two X Speeds and an X Accuracy for Seaking, then Red was finished. Nothing more to say!