It depends on your kid. My family thrives on structure. But we also do not believe in over-programming our kids so we try to strike a balance. So I think we typically had activities 3-4 days a week after school but made sure at least one was free (we had a nanny at that age). Plus often a sport on the weekends (I have boys). We were in a very good neighborhood public school.
It is a good age to expose your kid to new interests, socialize them, and make/strengthen friendships. But you also don't want to drive yourself nuts and break the bank. On the day(s) without activities they would often go to the local playground and usually classmates were there, but that is likely different for private where kids are more spread out.