Once you’ve selected a buddy and its profile icon appears in the bottom left corner of your screen, all you have to do to play with it is select your buddy’s profile. On your buddy’s profile, it’ll show different affection levels for things you can do with it, like walking, playing, or giving it treats.
To play with your buddy, select the “Play” option, and the AR feature with show your selected Pokémon in the real world around you. If you have the AR option deselected, your Pokémon will instead appear in a standard, grassy land.
All you have to do to play with your buddy is to pet it by stroking it on the screen. Do not tap or poke at your buddy, but instead rub it gently. You’ll know you’re doing this right if you see sparkles appear. While you’re in this mode, you can also give your buddy treats.
Doing this will complete the “Play with your buddy” task in the throwback challenge, but you should play with your buddy regularly anyway to increase its affection levels.
Don’t forget, for the throwback challenge, you’ll have to also feed your buddy three rare treats and take a snapshot — all of which can be done while you have the “Play” tab open for this task.
Source : https://www.distractify.com/p/how-to-play-with-your-buddy-pokemon-go