Some of the old moves are available to teach to your Pokémon. Here are a few of them and their locations [they will be taught by characters around these areas]:

Mega Kick/Mega Punch: Obtained outside of Mt. Moon, Cerulean Exit, by two men.

Seismic Toss: Obtained through the back entrance of Pewter Museum from a scientist.

Rock Slide: Obtained near the end of Rock Tunnel from a boy.

Softboiled: Obtained by a man across the pond in Celadon City.

Mimic: Obtained in Mimic Girl's house in Saffron City after you've given her a Pokédoll.

Thunder Wave: Obtained by a woman on the 2nd floor of the Sliph Co. building.

Counter: Obtained from a man located in Celadon's City Department Store 3rd floor.

Subtitute: Obtained from a man located infront of the Kangaskhan pen in Fuschia City.

Dream Eater: Obtained from a man across a pond in Viridian City.

Metronome: Obtained from a scientist in the second room of the Cinnibar Laboratory in Cinnibar Island.

Double-Edge: Obtained from a guy near the end of Victory Road.

Explosion: Obtained from a Hiker on the Northern area of Rainbow Island 1.

Elemental Attacks (Blast Burn, Frenzy Plant, and Hydro Cannon): Obtained from a move tutor in Rainbow Island 2. However, he will only teach your stater Pokémon one of these attacks according to its type.

Body Slam: Obtained from a tutor on Rainbow Island 4.

Swords Dance: Obtained from a man located on the pier of Rainbow Island 7.

Note: Some of these moves can be taught multiple times, others only once. So choose wisely.