Don’t use ゲームを遊びます for “play a game”
The following all works:
- ゲームで遊ぶ – using the means of the game to have a good time
- ゲームをする – do action on the game
- ゲームをやる – slightly more vulgar (I’m thinking of maybe more used by males?)
- ゲームをして遊ぶ – て form +遊ぶ meaning doing an action to achieve having a good time
Why
- 遊ぶ is 自動詞 (intransitive verb)
- Its meaning is “to have a good time doing things you like or are interested in”
て form + 遊ぶ
At first, I was confused about して遊ぶ, so I looked for more examples:
- シーソーに乗って遊びます – ride on seesaw (action) to enjoy the time -> playing on a seesaw
- 船に乗って遊ぶ – ride a boat (action) to enjoy or play -> playing on a boat
Reference