How do i record data of a game of tennis

Depends on the watch. On some models, there is a built-in activity profile for tennis, on others you may need to use a Connect IQ app, or record the session under another activity profile (for example Cardio, or just Other, or creating a custom profile for it). Once synced to the Garmin Connect account, you can edit the activity and change the activity type to Tennis there. 

Absolutely what trux said.

If you can tell us which watch you have, we can better assist you. There are some models, such as the Forerunner 955, Instinct 2, Fenix 7, epix (Gen 2), and Enduro 2, that offer tennis as a built in activity, but you might need to locate it on your watch. On my epix, Add >> Other >> Tennis