Pokémon-Amie is probably the absolute cutest thing in Omega Ruby/Alpha Sapphire - maybe the entire series. It could even be one of the most beneficial aspects for your Pokémon, since it can enhance your battle strategy and whatever personal attachments you have to your Pokémon.
The Pokémon-Amie app itself is accessed similarly to the PSS and Super Training apps on the Touch Screen via the PlayNav on the PokéNav Plus - simply press L/R to find it.
There, you can tap on a Pokémon. If you opt to switch it out, you can see its various stats regarding Pokémon-Amie and the other Pokémon with which you may want to play. Once you want to play, do so!
Within the Pokémon-Amie interactive, you can do a number of things.
Pet your Pokémon
You can use the stylus and Touch Screen to pet your Pokémon, which raises their affection. Keep in mind some Pokémon have areas they don't like to be rubbed on: for example, our Pikachu doesn't like being rubbed on his belly. It's generally individual to the Pokémon.
Some Pokémon have areas you outright shouldn't touch - people familiar with the anime can understand why you shouldn't touch Pikachu's cheeks, and general logic tells you not to rub the fiery tail of a Charmander.
Other than that type of stuff, your Pokémon is pretty okay with anywhere else; just rub repeatedly and a number of hearts (or a music note) should appear. The implications of this are in the next section.
Feed your Pokémon
You can feed your Pokémon PokéPuffs, by tapping the top-left icon in the interactive. From there, you can grab a PokéPuff - if you don't like what you see, maybe you should scroll left (put the stylus in the middle of the selection area and swipe left). Each Pokémon has its own individual likes - it disregards species.
There are several flavors of PokéPuffs - green ones are mint, orange ones are citrus, pink ones are sweet, light-brown ones are spicy, and dark-brown ones are mocha. There are also several degrees of effectiveness with each getting more powerful: basic, frosted, fancy, deluxe, and supreme.
Respectively, they give the Pokémon one, two, three, four, and five hearts - see the following statistics section for the purpose of them. The better you do in minigames, generally the better the PokéPuff you get. Feeding Pokémon PokéPuffs boosts their Fullness and Affection.
Quite literally, making faces at your Pokémon (or the 3DS/2DS). This does require a pretty bright area (we needed to be in a well-windowed area in the daytime) and a clean inner 3DS/2DS camera. When a certain green face-like icon appears in the lower-left, you can play this little minigame.
Basically, do what it tells you (wink this eye, tilt your head this way, open your mouth this much, etc.) and you can raise your Pokémon's affection! Keep in mind that your face needs to be pretty recognizable - no hair in the face, probably no glasses, and so on, like you would do for general facial recognition stuff.
You can run your stylus along the Touch Screen in areas where nothing are - it's basically to mimic you waving your finger in random motions. It doesn't really raise affection, but the reaction's pretty cute for some Pokémon, especially if you high-five them. It doesn't work for all Pokémon, though.
Similarly, you can use the microphone to speak with your Pokémon. (The mic will take just about any random sound into account, though. It also doesn't really change anything.
As you play with your Pokémon, you will occasionally earn decorations to pretty up the play area on the bottom screen. The decorations sadly don't follow into the actual Pokémon-Amie play area, but it's a nice option all the same.
You can also receive decorations (and Poké Puffs) from visiting Pokémon when you have Pokémon-Amie as your active app while connected online.
You can also play minigames with them, discussed in later sections - basically, they raise Enjoyment and Affection, as well as lowering Fullness. Think of it as a healthy workout for your Pokémon.