Depending on the game, raising a Pokemon's happiness can be easier or harder. The more recent a game is, the easier it tends to be.
The literal fastest way to max a Pokemon's happiness is to follow these steps:
Give your Pokemon a Soothe Bell. This will increase the amount of happiness a Pokemon will gain from any given happiness-raising activity by 50%.
Go to your Pokemon's met location. This will increase any friendship raise by 1, as long as you're not in Gen 4.
Feed your Pokemon ~18-20 happiness raising berries. Depending on how much you've used this Pokemon and what it's current happiness is, you may be able to use less. The berries that raise happiness are the Grepa, Hondew, Kelpsy, Pomeg, Qualot, and Tamato. Starting at a happiness level of 0, 18-20 berries will increase your happiness by about 200 points. Once your happiness is at 200 points, happiness berries become obsolete, however you only need 20 more points until you reach the required amount for evolution. This can be obtained in a number of ways.
The easiest way is to go get a massage or spam Rare Candies. A massage at minimum raises your friendship 5 points, and can sometimes raise it 30 points. Rare Candies, (prior to SwSh,) can raise your happiness 2-3 points for every level gained. After combining these two factors, you should reach the 220 Friendship threshold quickly.
As you can see, there are a lot of ways to raise friendship in Pokemon. The simplest way is to slap a Soothe Bell on your Pokemon and battle with it until it evolves. The berry method is a lot harder to accomplish in some games than than in others, so feel free to explore other options like leveling up, walking around, and playing with your Pokemon.
BIG help if you're looking for ideas: Source
Let me know if there's anything I forgot or overlooked, there were a lot of ideas that went through my head coming up with this answer. Hope I could help!
