Use a Lucky Egg either before an evolution-fest (as described in page 2 of this article), before going to a lure module-filled location, or before doing the two gym training tricks as described in the previous page. Do this for extremely quick level-ups and make the most of your Lucky Egg, as they are fairly rare.
One of the great debacles of all time is whether you can get a specific Eevee based on what moves they know, or whether the evolution is just completely random. The going theory is that if it knows Dig, you get Flareon. If it knows body slam you get Jolteon. If it knows Swift you get Vaporeon. However, that has not been fully reliable, so the best word right now is that the evolutions are completely random.
Almost everybody has it wrong when it comes to trying to get a “Nice!” bonus. The way to get one is to stop the circles and throw the Pokéball when the colored circle is completely overlapping with the white circle. That will get you the bonus and give you the largest chance of catching the Pokémon.
If you are in the middle of a Pokémon fight and the Pokéball stops bouncing or the game freezes, close out of the game and then open it back up and go to your journal. The journal will tell you if you caught it or if it ran away. If you did catch it, then you can also go to your list of Pokémon and if the list shows the latest first, the new Pokemon will show up on your list.
If you are playing the game on an Android device, you can speed up egg hatching by going to the Location Mode area of your settings and changing it to Battery Mode. This will cause your location on the map to bounce around a lot, and not only will this save your battery life, but you’ll hatch eggs far more quickly, and you won’t even have to move around to make it happen.
Each time that you level up, the max CP increases for your Pokémon. Fighting more gym battles and training at a gym will not increase the maximum CP of your Pokémon, but often it can seem that way because of the game’s rampant glitch that causes you to not be notified when you level up sometimes.