Ways to induce happiness:
- Walk 256 with the Pokemon in your active Party. (+1 for all happiness values)
- Take the Pokemon to Blue's Sister (Daisy) in Pallet Town for Grooming from 3 PM to 4 PM everyday. (+3 for values 0-199, +1 for values 200-254)
- Give the Pokemon Vitamins [HP Up, Carbos, Iron, Protein, Calcium, PP Up] (+5 0-99, +3 100-199, +2 200-249)
- Usage in Important Battle [Gym Leaders, Elites, Red] (+3 0-199, +2 200-249)
- Level Up the Pokemon (+5 0-99, +3 100-199, +2 200-249 per level)
If you want to skip all this happiness stuff and evolve the appropriate Pokemon much faster, you can get a special PokeBall made in Azelea Town by giving Kurt a Green Apricorn (these can be gotten daily from Cutting down the bush on Route 42 and checking the Apricorn trees.) If you catch any Pokemon with this ball, it starts with a happiness value of 200. Unfortunately, Chansey and Golbat are the only catchable Pokemon that evolve from happiness.
So if you want to evolve your Eevee into, say, an Umbreon, the quickest way will probably be to cycle around with it in your party and use it in Gym battles.