i would get them in the ground. with good soil, mulching them and watering they should be fine. you can also make a small canopy to shade the seedlings.

Agreed, put them in the soil ASAP. They grow some killer roots to support growing those big fruits, so you'll stunt them pretty easily if you leave them in pots for more than a week or two after they sprout.

If there is any doubt about the heat, then use some shade cloth. You can buy the expensive real stuff, or you can just hit the hardware store for some screen door mesh. Its basically the exact same thing, but far, far cheaper.

GOOD LUCK! I am in Orange County Ca., and you will have fun with your kids with this project. There is a terrific seed starter soil, I used it this year. You can go to Home Depot and get the starter sets! But get them growing now! The weather has been GREAT!