I'm trying to breed an Eevee with perfect IVs that I will later evolve to a Vaporeon. I know how to breed but I'm not sure if there is another factor that I have to look at while leveling it. One of the things is the usage of the water stone. Should I use it right at level 1 or should I wait untill my pokemon reaches a certain level? Does the time of the Evolution have a positive or negative impact on the pokemons stats? Also Are there some other small things that I have to keep in mind while leveling it?
The Evolution level only matters if you are holding off to get a specific moves, which in Eevee's case probably doesn't matter that much, unless you want Take Down or Double Edge.
It doesn't impact stats at all. In other words, whether you choose to evolve at level 1 or level 100, they will have the same stats if you compare them both at the same level once both are evolved (assuming IVs and EVs are the same).
In case you were wondering, the Move Reminder can help if you're in a bit of a 'Catch-22' situation.
Say you want to wait until Eevee learns Double Edge (37), but also want to evolve earlier so that Vaporeon learns Acid Armour (29). What do you do? You wait until Eevee learns Double Edge, and then evolve it. Take it to the Move Reminder, who will be able to teach it Acid Armour.
It doesn't work in the reverse scenario:
One limitation is the Move Reminder is restricted to the particular moveset associated with the Pokémon's species (i.e. current evolutionary form). Many Pokémon, especially those that evolve by way of evolutionary stone, have access to moves that their evolutions do not, such as with Eevee, who can learn Take Down by level though none of its evolutions can.
Emphasis mine. Source