With eight separate evolutions, you might think that getting all of them is a little daunting. Fear not, the original three evolutions are practically handed to you if you explore each area carefully and find each of the necessary Stones. Sylveon is rather easy to get as well, as long as you don't mind spending some time in Pokemon-Amie. Glaceon and Leafeon are perhaps one of the easiest evolutions, though they do require you to progress through the game a bit.
Glaceon
In order to get Glaceon, you'll have to travel to Dendemille Town and go to the Frost Cavern. You'll also need Surf. In there, just climb down a few flights of stairs until you reach the only stream in the cave. Surf across it and you'll find a pathway down which will lead you to a single room with a Glacier in the middle of it. Simply walk around until you get into enough battles to level Eevee up and voila, Glaceon! You can also speed this process up by having a Rare Candy and using it right next to the Glacier which will immediately evolve Eevee into Glaceon.
Leafeon
Leafeon is even easier to acquire, since he doesn't actually involve venturing deep into any cave. The only caveat is that you can't get Leafeon until you reach Snowbelle City, which houses the last Gym in the game. From Snowbelle City, you have to venture off onto Route 20 the Winding Woods, which is also a part of your main questline. Upon entering the Winding Woods, you'll come across a stone that's surround by a patch of grass (resembling a Pokeball). Touch the stone so you get the message "The surface is covered with moss. Touching it feels good somehow." Once you get that message, either battle to level Eevee up or simply use a Rare Candy and you'll get Leafeon.