To participate in many activities can be expressed in Engpsh as "to be involved in numerous activities" or "to be active in a variety of events." For example:

I have been involved in numerous activities this semester, including club meetings, volunteering, and sports practices.
She is always active in a variety of events, from charity fundraisers to art exhibits.

You can also use the phrase "to be busy with" to describe being occupied with many activities:

I have been so busy with work and school that I haven't had much free time.
He has been busy with meetings and projects for the past few weeks.

You can also use the phrase "to have a full schedule" to describe being occupied with many activities:

My schedule is full this week with classes, work, and a few social events.
She has a full schedule with rehearsals and performances for the upcoming play.