Growlith is a prime example. Arcanine only learns extreme speed at level 49! You don't want to train a Arcanine that only knows bite do you?
It learns ember in early levels and even flame wheel at level 31 . At level 49, it learns flamethrower! It's worth to train Growlith to at least level 49 before evolving it. There are plenty of other good Pokémon and Milktank isn't even that good. It's just bulky.

Same goes for Vulpix. Ninetail doesn't learn anything until level 45. It will learn Fire spin.. A 15 base power move with 75 accuracy <.<.
Train your Vulpix! It already starts with ember, so that's a good call. It doesn't really learn anything good till level 29, but at level 29 it learns flamethrower! At lv17 it learns will-o-wisp. Not too good considering it only has 70 Accuracy, but the burn status is useful since it cuts your opponent's attack!

I don't understand why people would waste their Eevee on an fire stone. I mean, use it for something else xD.
Eve has Umbreon and Espeon that it can evolve in! I personally think it's a waste to evolve Eevee into Flareon, since you can already catch Flareon near slateport.Flareon might just be the only one of the starter evolutions that actually has a move pool, but learns flamethrower at level 52.. While growlith learns it at level 49 and vulpix at level 29..
If you really want to evolve Eevee, then I would suggest evolving it into Espeon (move pool wise) Espeon has a better move pool then Umbreon. Umbreon  does learn faint attack at level 36, which is 60 base power and doesn't require accuracy, but Espeon learns Psybeam at the same level having a base power of 65. Espeon learns confusion at level 16, which is the first actual psychic move that it learns. And at last, Espeon learns Psychic at level 47.  since you can already catch Vaporeon and Jolteon in the wild, I don't see any reason to evolve your Eevee into one of these. But if you're planning on evolving it in either one of them then it's up to you honestly. Jolteon doesn't learn thunderbolt, by leveling but Vaporeon does learn aurora beam. I know, they're not related, but the best move Jolteon learns before reaching  level 52(thunder) is Thundershock.. While Vaporeon learns an pretty useful move, Acid armour. It has 40 PP but sharply increases defence. That's perfect for a bulky tank Vaporeon.

I hope this information was useful, good luck with your game!

-Yngwie-

Yngwie : This info is good to know. Thanks for posting it. I might start playing this soon

Thanks for the guide. 

Okay I'm gonna assume this is a Gen 3 hack, so: Growlithe is a physical pokemon, so it does not want Flame Wheel or Flamethrower, both are special moves. He might want Flame Wheel but ES is better for him. And Umbreon has a niche as a tank and Dark isn't as common of a type. so umbreon is still useful.

woohoo982 : Actually, Growlithe and Arcanine are mixed attackers. Growlithe's special attack is the same as its physical attack and Arcanine's special attack is only 10 points lower than it's physical attack (but special attack is still a base of 100). Plus it gets STAB from flame wheel / flamethrower which makes them even more powerful. They're both better options than ES, which only has 5 PP.

This is good for some people, but personally I don't care as much for the attacks as for the actual stats of the Pokémon. I'd much rather level up an evolved Pokemon and get the evolved stat boosts than train an unevolved one until I can learn the moves because with some Pokemon, the unevolved ones will gain stats differently and most likely would not be as strong as the way I want them, even with the same moves.