You will hopefully enjoy the summer here - less hot in temperature but more 'hot' in humidity - I mention this to keep in mind as you are looking at summer camps.
So, the nice thing is that it is usually coolest in the mornings and warms up throughout the day. hit the playgrounds at the parks early - then move on to a pool, then inside as it gets hotter!
I don't live in Cary but I googled the parks and rec link for you:
Programs and Classes
I live in the Raleigh and love the programs there throughout the years for my kids so parks and rec offerings are definitely worth looking into.
Lots of VBS around here -- the churches all have signs up and some have them listed on their websites. It is worth a google search for churches close to your location and then making some phone calls to find out when their VBS is scheduled. There are so many around here, you could spend your summer doing VBS! Some have VBS during day and some in the evening hours. So make sure you ask the time as well as the dates.
As far as museums - there are loads of free things to do - NC Museum of Art, NC Museum of Science as well as 'pay for' the Museum of Life & Science in Durham and Marbles Kids Museum in Raleigh. Then, you have the planetarium in chapel hill which is fun to do at any age.
If you want to drop $60-100, then you might consider 'frankies' off of Alexander Drive and Rte 70 in Raleigh - It gets pricey quickly so get ready to set limits before you go so it doesn't cost you a fortune!
Best of luck on your move.