Which of the original 151 Pokemon learn moves AS they are evolving?

Question

In Pokemon Red and Blue, which Pokemon learn moves at the same level at which they evolve?

or

Which Pokemon (limited to the original 151) learn moves as they simultaneously evolve?

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Originally this question was asked somewhere else: https://gaming.stackexchange.com/questions/27854 however, the original asker deleted it with in 11 seconds of me trying to post the answer. Since I spent the time to work out the answer, I wanted to preserve the question.

Answer 1

Now, I might be wrong, but my understanding is that no Pokemon learns moves during the act of evolution. All moves are learned through: breeding, leveling, or HM/TM.

That said, there are Pokemon who evolve at levels where they also learn moves. The two mechanisms are disjoint, but can occur at the same time.

For example: Charmander, learns Dragon Rage at level 16. It also evolves to Charmeleon at level 16. Thus it can learn Dragon Rage and then evolve after a single battle.

Assuming you want a list of these occurrences the answer is:

Before Evolving

• Charmander to Charmeleon (level 16, learns Dragon Rage)
• Squirtal to Wartortle (level 16, learns Bite)
• Wartortle to Blastoise (level 36, learns Skull Bash)
• Poliwag to Poliwirl (level 25, learns Bubble Beam)
• Geodude to Graveler (level 25, learns Smack Down)
• Magnemite to Magneton (level 30, learns Lock-On)

After Evolving

• Bublbasaur to Venusaur (level 32, learns Petal Dance)
• Charmeleon to Charrizard (leavel 36, learns Wing Attack)
• Metapod to Butterfree (level 10, learns Confusion)
• Kakuna to Beedrill (level 10, learns Furry Attack)
• Rattata to Raticate (level 20, learns Scary Face)
• Ekens to Arbok (level 22, learns Crunch)
• Sandshrew to Sandslash (level 22, learns Crushclaw)
• Oddish to Gloom (level 21, learns Mega Drain)
• Venonat to Venomoth (level 31, learns Gust)
• Digglet to Dugtrio (level 26, learns Mud Bomb)
• Mankey to Primape (level 28, learns Rage)
• Abra to Kadabra (level 16, learns Confusion)
• Ponyta to Rapidash (level 40, learns Fury Attack)
• Slowpoke to Slowbrow (level 37, learns Withdraw)
• Seel to Dewgong (level 34, learns Sheer Cold)
• Gastly to Haunter (level 25, learns Shadow Punch)
• Rhyhorn to Rhydon (level 42, learns Hammer Arm)
• Magikarp to Gyarados (level 20, learns Bite)
• Omanyte to Omstar (level 40, learns Spike Cannon)
• Kabuto to Kabutops (level 40, learns Slash)
• Dragonair to Dragonite (level 55, learns Wing Attack)

Answer 2

It appears the question was about Pokémon Red, Blue, and Yellow specifically. Almost every Pokémon had their movepools and the levels they learnt moves changed from Gen I to the Advanced Generation (Gen III), so here is a list of the Pokémon that learnt moves as they levelled up within the Gen I games (solely Red, Blue, and Yellow):

• Charmeleon to Charizard, Level 36, Wing Attack
• Caterpie to Metapod, Level 7, Harden (only Yellow, not Red or Blue)
• Metapod to Butterfree, Level 10, Confusion (only Yellow, not Red or Blue)
• Weedle to Kakuna, Level 7, Harden (only Yellow, not Red or Blue)
• Mankey to Primape, Level 28, Rage (only Yellow, not Red or Blue)
• Poliwag to Poliwhirl, Level 25, Doubleslap
• Bellsprout to Weepinbell, Level 21, Stun Spore
• Slowpoke to Slowbro, Level 37, Withdraw
• Magikarp to Gyarados, Level 20, Bite
• Dratini to Dragonair, Level 30, Slam

_{(Information compiled from Serebii.net's RBYGSC Pokédex).}