打游戏 is the everyday verb for playing video games — for any platform: PC, console, mobile. More specific verbs:
玩游戏 (also 'play games', slightly broader, includes board games),
上号 (shàng hào, 'log onto a game account'),
开黑 (kāi hēi, 'play in a voice-chat squad with friends').
打 here suggests active engagement/combat.
