General:
* Gaming
* Playing
* Engaging in recreational activities
* Having fun
* Amusing oneself
Specific to board games:
* Partaking in board games
* Playing tabletop games
* Strategizing
* Challenging oneself
Specific to video games:
* Gaming
* Playing video games
* Leveling up
* Exploring virtual worlds
Informal:
* Hanging out
* Chilling
* Having a blast
* Dicking around (casual, potentially slang)
More formal:
* Participating in competitive activities
* Engaging in strategic thinking
* Developing problem-solving skills
The best word to use will depend on the specific context.