I play the original Silver mostly, but I've dabbled in the original Yellow (haven't done much with Blue after a friend just returned it, and I've lost Red.)
Lesee here..
for Yellow:
Mr. Mime- best psychic in generation 1. He defends with barrier to keep himself alive, and attacks with Confusion and Thunder Bolt, and may paralyze the foe with a Thunder Wave. This guy generally takes out the physical attackers and ghost-types, but may substitute for Pikachu with electric attacks, and paralysis never hurt nobody.
Blastoise (Named him Battletoad for whatever reason)- This guy is my murder machine when it comes to electrics, fires, grounds, rocks, and Lance. Surf, Ice Beam, Dig, and Strength are his attacks, and almost nothing can survive the onslaught of attacks.
Nidoking- Nidoking is my brute force. Earthquake kills off electrics and rocks, Bubblebeam repels fire, rock, and ground, Thrash helps when nothing else is available, and Blizzard murders all. Only a lucky Fissure can fell my Nidoking.
Pikachu- well, it's Pokemon Yellow. You shouldn't be "shocked". Thunder Bolt and Thunder lay the smack down on everything but ground-types and dragons. Thunder wave holds everything down, and pay day rakes in cash in wild battles.
Charizard- With a quick Slash, most non-rock and non-ground types will suffer quite a bit. Flamethrower and Fire Blast demolish Ice-types, Grass-types, Flying-types, and Fighting-types. Fly can pick off any Poison-types and evade a hit.
Dragonair (soon to be evolved)- Thunder, Blizzard, Fire Blast, and Surf. All four main elements help destroy anything that isn't capable of a quick STAB ice-attack.
On Silver I generally use...
Heracross- Megahorn rules over the Psychics and Darks, Rock Smash crushes physical typed pokemon, Reversal definitely helps in times of peril, and Earthquake just about murders everything else.
Tyrranitar- Raised from an egg, this monster wields Crunch to scare off Psychics, Rock Throw to knock down aerial opponents, Earthquake to replace Heracross if necessary, and if all else fails, Hyper Beam finishes off most other non-mineral types.
Alakazam- Recover, Psychic, Fire Punch, and Ice Punch make a deadly combination. Recover can quickly heal up, Psychic whomps everything but Darks and Steels, and the elemental punches just kill for the sake of type-advantages.
Gengar- Hypnosis rules with an iron fist, alongside its partner moves Nightmare and Dream Eater, which can kill anything in only 4 turns; even Red's whole team. Shadow Ball holds no real merit, but it's cool to have.
Muk- This here is my best warrior for almost everything. Fire Blast, Ice Beam, and Thunder reign supreme among almost every type, and for those that resist any of those attacks, Explosion usually finishes them off pretty well.
Snorlax (whom I named Lyra out of insult towards the character)- The wall. Body slam may paralyze, Rest heals up, Snore fights during rests, and Surf gets rid of anything that could break its defenses.
I hate conforming to the usual smogon tiers, especially when people claim OU and UBER are the best, but I've found ways around them.