Summarize:

This is not required for the trick to work, but will allow you to level up twice as fast. To purchase a Lucky Egg, tap on the PokeBall on the main screen and tap the "Shop" icon. Scroll to the third row of items to find Lucky Eggs. These Pokemon are the easiest to level up, which is how you will gain your experience using this trick. Pidgey is the best Pokemon to use for this reason, but Weedle and Rattata work as well. In return, he will give you candy, which you can use to level up your Pokemon. To transfer Pokemon to Professor Willow: Tap the PokeBall on screen, and then the Pokemon symbol to open your list of Pokemon. Tap on a Pokemon that you have a duplicate of, and hit the "Transfer" button. This will open a list of the Pokemon you have caught. Tap on a Pokemon you have caught multiple of, and hit the "Transfer" button. Professor Willow will take the Pokemon to his lab, and give you candy in return. This candy can be used to level up and evolve your Pokemon. From the main screen, tap the PokeBall near the bottom and then the "Items" button to open up a list of all items in your pack. Tap on the Lucky Egg you have purchased to use it. This will grant you a 30-minute period during which all of the experience you gain will be doubled. Since this period only lasts 30 minutes, be sure to complete the rest of this method once you have enacted this step! Your Pokemon will level up very quickly, causing you to gain a huge spike in XP and, depending on your current level as a trainer, allow you to increase by a level or two.The evolution animation takes almost 30 seconds to complete, meaning that you could get about 60,000 to 65,000 XP if you use up the entire 30 minutes on evolving. You're well on your way to becoming a stronger trainer!
Buy a Lucky Egg (optional). Catch several weak Pokemon of the same type. Transfer your duplicate Pokemon to Professor Willow. Tap the Pokemon button. Transfer your duplicate Pokemon to Professor Willow. Open up your Items. Use your Lucky Egg. Use all of your candy on the Pidgeys in your pack. Repeat these steps over to gain more experience.