Instructions
Travel to Ecruteak City. It's west on Route 42, east on Route 38 and north on Route 37.
Enter the Pokemon Center, which is on the south side of town. As you enter, you'll have a conversation with Bill, the man who programs the PC Storage System. He invites you to his house in Goldenrod.
Go to Goldenrod, which is north on Route 34 and south on Route 35. Enter Bill's house on the south side of town. Speak with Bill and he'll give you an Eevee.
Put the Eevee in your active party. Your goal is to make the Eevee happy. In order to do this, always keep it in your party during the day. Every 256 steps you take increases its happiness by 1 point. When it gains a level, its happiness goes up by an additional 2 to 6 points, so also let it fight battles. If it faints or if you have to use restorative items like Heal Powder or Energy Powder, it loses happiness.
Take the Eevee to Veilstone City, which is east on Route 215 and north on Route 214. Once a day, you can get a massage at the Massage Parlor next to the Gym. Massages increase Eevee's happiness by another 3 points.
Remove Eevee from your party at night. If it levels up during the night, it might turn into an Umbreon.