Well, every family is going to be different on it - but I do think you need to prioritize - and probably place limits - especially if you have a few kids. There is only so much time - and for that matter, money.
Here are my family's priorities.
We avoid sports... I'm sure there is a lot of value in them, but they take a lot of time (and for some, time all over the calandar - ie, variable on days etc.)
We do Swimming. We consider this an important Safety skill. If they do well at it, it can also be a good part-time job when they are a teenager. (my neices and nephew all became life guards. My one neice also became a SCUBA instructor - and, along with other skills, worked full time at a few club-meds because of this skill.) But hey - safety around the water is my main goal.
We also do Music lessons. I've always felt it was an important skill and talent. We did Suzuki Violin partly because I was familiar with it and liked its ideas - and partly because when my son was 3 he wanted a violin.
We started both of those young, at age 3.... although most people wait for music lessons unless they are doing Suzuki.
At 8 our kids start Scouting. Well, that is when the oldest started (this year) - but we ended up starting the next in the younger program at almost-6 because she wanted to do it to. First, Scouting is the program for boys in our church (although in Canada scouts is co-ed outside of the church...) But also... well, I've been a Scouting (or Girl Guide... for a few years) leader for over 20 years now. My husband has even more experience. So let's just say we really value the MANY benefits of a well-run scouting program.
And that is it. That is the 3 things we do.
Well...... when our kids are 12, if they want to join Cadets, they can. (in Canada there is Army Cadets, Air Cadets, and Sea Cadets... sponsored by the military) Both their Dad and I were in Air Cadets, and we both joined the military afterwards. He was a Tank Commander. I was to be an Aeronautical Engineer (did you know that Officer Cadet Training (ie, Basic training) will almost certainly expose any tendancy towards depression... much more than the Air Cadet camps will???) So anyway, if they want to be an Air Cadet, or an Army Cadet, that is fine with us. (Although if they want to be an Army Cadet, I think their dad would insist on them doing it in the group associated with his Regiment.) If they want to be a Sea Cadet........ hm...... that might be a harder sell (lol)
Anyway - that is our priorities. The kids would have to be pretty convincing for other activities - although I'm not saying they wouldn't be allowed.... but one thing we discovered (well, my dh already knew as he has 3 adult kids too) - generally if you let one child do an activity... the others want to do it too. That can get expensive really fast. Oh, and driving kids to various activities every evening wouldn't be much fun either.... especially when the activities collide.