Normal TypeEevee is normal type, which can be fairly tricky to use properly. Normal type Pokemon do not have the advantage against any other types, and they are weak to fighting types. Normal type moves cannot harm a Ghost type, and Ghost type moves cannot harm Normal types. There are no easy advantages to be found here. How well your Pokemon does is going to have a lot to do with its base stats, as well as its nature. Eevee's base stats really aren't so great though, and nature can only help but so much. You can love Eevee to bits like me, but it won't do particularly well on the battlefield (especially in competitive).
According to Bulbapedia, "Normal-type Pokemon, on average, have the lowest Defense and Special Defense of all Pokemon and of fully evolved Pokemon". Normal type Pokemon are not gifted in the realm of defense, and generally are not utilized for this purpose. Normal types suffer a similar situation in regards to attacking, but because of the wide range of non-Normal type moves that can be learned, many times weaknesses can be covered. Using Eevee competitively is not recommended, even in the 'little cup' of unevolved Pokemon. Smogon recommends that if you must use Eevee, using the 'Wish Passer' strategy is best. You can find details of that here.
Through normal leveling up, Eevee does in fact learn a couple of moves that are not Normal type. Eevee learns the Ground move: Sand Attack, the Fairy moves: Baby-Doll Eyes and Charm, and the Dark move: Bite. Through the use of TMs, Eevee can also learn certain Poison, Fire, Water, Ghost, and Psychic moves. Breeding can also play a part in learning different move types. For more information about Eevee's learnset, please refer to the learnset page.
But if you don't particularly mind these problems and don't play competitively, you can still use your Eevee for sure! I almost always have an Eevee on my team. I honestly can't recommend any particular moveset (I never make note of them), it's just a lot of trial and error. Having Eevee around in the later generations (X/Y and forward) provides the ever-fun benefit of being able to play with your Eevee.
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