Many Pokémon evolve through leveling, but not with all.  For example trading with another trainer or a special item to make it evolve.  Each type of Pokémon specie has their types of abilities.  For example, a pokémon learns swift at lvl 11 and will learn defense curl at lvl 13 but evolved using a moon stone before they reached lvl 13, it will not learn it.  Only if you use TMs and HMs.

Here are some Pokémon that need to be evolved using trade or special items.

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Special Stone Evolution

The following Pokémon could be evolved using a special stone.  They're 5 types of stones, Fire, Thunder, Water, Leaf, and Moon.  The fire, thunder, water, and leaf could be bought as the Celadon City mart, while the Moon stone could only be found all throughout the game.

Fire Stone

Vulpix#37 - Ninetales#38
Growlithe#58 - Arcanine#59

Thunder Stone

Pikachu#25 - Raichu#26

Water Stone

Poliwhirl#61 - Poliwrath #62
Shellder#90 - Cloyster#91
Staryu#120 - Starmie#121

Leaf Stone

Gloom#22 - Vileplume#23
Weepinbell#70 - Victreebell#71
Exeggcute#102 - Exeggutor#103

Moon Stone

Nidorina#30 - Nidoqueen#31
Nidorino#33 - Nidoking#34
Clefairy#35 - Clefable#36
Jigglypuff#39 - Wigglytuff#40

There is only one particular Pokémon that could be evolved using all three(fire, thunder water), which is Eevee.  Eevee could evolve into Vaporeon, using water stone, Jolteon, using Thunder stone, and Flareon, using fire stone.  But becareful, you could only change it once and to only one type!  There's no turning back on it!  So, choose wisely...

Evolution through Trade

The following Pokémon can only evolve through trade.  After the Pokémon has been traded, the Pokémon immediately evolve right after!

Kadabra#64 - Alakazam#65
Graveler#75 - Golem#76
Machoke#67 - Machamp#68
Haunter#93 - Gengar#94

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Weedle will bring you back to the main entrance!  Becareful, don't step on him!!