Are there any places near Manali where we can go for trekking and camping together as a family (kids, wife, and me)?

I would say yes there are lot of places near Manali where you can go for trekking and camping together as a family.

For Trekking you can go for Jogni Waterfalls with your kids and wife and for Camping you can do it in Old Manali.

And if you want to do both together and also want to go for a beginner trek with your family you must plan for Bijli Mahadev Trek where you can go for trekking easily with your family and even kids can do the trek easily and also you can go for Camping at Bijli Mahadev only.

Bijli Mahadev is nestled between Kullu & Parvati Valley and at top of the Kharahal Valley which makes it the perfect spot for Trekking & Camping with family and kids as well. You also get an eagle eye view from there which makes the place a must visit in Kullu with mesmerizing snowcapped mountains.