How do you eve Pokemon fast?

Answer 1

"Eve", I suppose, can mean one of two terms when relating to Pokemon: Evolving, and EV-training.

Most Pokemon evolve by leveling up to a certain level (eg. Tentacool evolves into Tentacruel at level 32; Magikarp evolves into Gyarados at level 20; etc.). For these Pokemon, you find an area where the Pokemon give excellent EXP, such as at Meteor Falls or Victory Road. A Lucky Egg increases the overall EXP yield a Pokemon receives after battle (Lucky Eggs are held by wild Chanseys). An EXP. Share allots EXP to the Pokemon holding it, even if that Pokemon didn't participate in battle, and an EXP. All allots EXP to every Pokemon in the party (However, EXP. All was only available in Red/Blue/Yellow). Rare Candies also increase a Pokemon's level without it having to fight, and Rare Candies are easily gathered using a Zigzagoon's Pick Up ability.

A few other Pokemon can only be evolved by using an evolution stone (eg. Pikachu evolves into Riachu when exposed to a Thunder Stone; Clefairy evolves into Clefable when exposed to a Moon Stone.)

And a few others are evolved when traded between to games (eg. Haunter into Gengar, or Kadabra into Alakazam). Some even have to be holding a certain item to evolve when traded (eg. A Slowpoke evolves into a Slowking when traded while holding a King's Rock).

A few more evolve when leveled up with high happiness, but these are mostly baby Pokemon (eg. a happy Azurill evolves into Marill; a happy Munchlax evolves into a Snorelax).

As for EV-training, if a Pokemon is holding a Macho Brace, it receives twice the effort values it would receive normally. Since you asked for only the 3rd Gen games, I won't delve into all the 4th Gen equipment stuff, since it isn't pertinent to the question.