All of them. Just get a full team of eeveelutions. Rotate out a couple as needed, but just put together a full assortment of eeveelutions. You will regret nothing.
Edit: I have just been informed by my editor that this is “an inadequate response” and that I should “take legitimate questions seriously, Bill,” so allow me to revise the above statement.
Let’s assume that you’re either starting with an eeveelution of your choice or that you’re starting with an eevee that you have the option of evolving using any means possible soon after you receive it. (We can also assume that “any means possible” includes the method needed to obtain a sylveon, which evolves via a technique most easily performed in Kalos.) Let’s also assume that you intend on tackling the traditional order of the Sinnoh League circuit, which begins in Oreburgh and ends in Sunyshore.
That all said, which you choose depends entirely on your battling style and what the rest of your team will be by the time you reach Eterna. If you’re the sort who battles according to type advantages, the very first gym in the circuit is run by Roark, who is a rock-type specialist. Therefore, the easiest eeveelutions to have with you would be vaporeon, leafeon, or possibly glaceon, whereas the most challenging eeveelutions to have would be flareon and jolteon. However, the second gym is run by grass-type specialist Gardenia, which means glaceon will have the overall best advantage for the first two gyms, followed by leafeon (which has a defensive advantage against its own type).
By the time you clear these gyms, you should gather together enough pokémon that the rest of the circuit will depend less on what type your eevee has become, although it certainly doesn’t hurt to take these into consideration. Which is to say after the second gym, glaceon may struggle with Maylene unless you’ve obtained support in the form of a psychic- or flying-type (of which there are numerous options in Sinnoh), as well as Byron (who should really be approached with fighting-types anyway). Vaporeon will face far more struggles thanks to Pastoria’s water-type gym and Sunyshore’s electric-type gym, and leafeon should be brought nowhere near Snowpoint’s ice-type gym. Again, though, all of these have counters in the form of Sinnoh’s native pokémon.
On the other hand, if we consider average battling capabilities, espeon has an excellent distribution of speed, offensive, and defensive abilities, making it one of the easier eeveelutions to train. Taking into consideration the fact that its type is neutral to the first two gyms and effective against the third, espeon may be your best choice, but you’ll encounter difficulties against ghost-type specialist Fantina, as well as against Elite Four members Lucian and Aaron. Alternatively, there’s jolteon, which is slightly faster than espeon, but it’s slightly weaker, it lacks physical defenses, and it will run into trouble with not only Roark’s partial-ground-types but also Byron’s, as well as Volkner’s electric-types.
In my personal opinion, however, it’s much better to go with a more defensive pokémon if you’re starting out than one that can hit quicker and faster, which means either flareon or umbreon would be ideal choices. It should be noted, though, that although flareon excels in defending against special attacks, it does not do so well in defending against physical ones, which is a problem when going up against Roark’s gym. However, if trained well, it would be an effective team member, especially in light of the fact that fire-types are rare in the Sinnoh region, whereas many of the obstacles you may face might require the assistance of one. Meanwhile, while umbreon has the best overall defenses of any of the eeveelutions (never mind type advantages against both Fantina and Lucian), it’s also the slowest and weakest offensively. However, umbreon’s defenses are so impressive that the trick with training it is simply to chip away at an opponent while they struggle to defeat you.
As for sylveon, it’s not exactly a terrible choice at all, but it should be noted that a fairy-type would struggle against Byron, Flint, and part of Aaron’s team. On the other hand, it would excel against Maylene, and it boasts decent—but not necessarily excellent—offensive and defensive capabilities (but rather unimpressive speed).
In short, for a beginner, it depends. If you focus on type advantages, leafeon may be the best choice, all things considered. Espeon and flareon are the best for those interested in offensive pokémon, and umbreon is the best in terms of defensive trainers. Get a jolteon if you wish for something speedy.
But really, all of them are excellent choices.
The Pokédex According to Bill 