Let me say right up front in this one--I am 60 years old, my only living child will be 34 this summer and is not married yet. I have no grandchildren. I do have several nieces and nephews, none who live close enough to do these things with. But, I have taught school at various levels for many many years, so I do know kids and I was one once.
1. Go to the public library. When I was a child and my dad was working on a degree, I got to go to the library with him every week while he studied. I am sure, absolutely sure that is how I became a reader. That is why I have always sought out the public libraries in the various places we lived. I took my son to the library. Despite the fact that he apparently loved the maths more and became an engineer, he is to this day a reader of books. Go to the library and spend time there.
And read out loud to your kids. (And, yes, I am a librarian.)
2. Go to the park. Every place has a park. Parks don't have television or game system and you meet other people. Scope out the good parks, figure out a good time to go, and go. Even better, arrange to meet other moms there.
3. Do something with water. This can be running through the sprinkler, going to the pool, an inflatable pool, water pistols. Water and summer go together. Do something with water.
4. Plan a craft or an art project or science-y or social studies something. This could range from water color painting, to rubber band bracelets to a garden to learning a language. I did a lot of these things with my son, but looking back, I wish I had done even more of this.
5. Look into the free or cheap stay-cation possibilities in your area. I live in a town with a free zoo, science center and history museum. Do some of that. Even if you are going to camp or on a real vacation, learn something about your area.
My point is--do something! Be the leader and the motivator. It is all too easy to stay home and do nothing. That will not make memories, forge relationships, or build character. Don't ask the kid what he or she wants to do. They don't even know the possibilities until you have planned and they have experienced some of these things. And I am not talking planning and doing something every minute of every day. Find a balance based on the numbers and ages of your children.