I would highly recommend you looking into New Smyrna Beach, it is just south of Daytona and is a much more quiet, typical, beach community. Like Daytona, you can drive on the beach -which the kids will probably think is pretty cool(I know they have restricted access since I lived in the area, but I believe there are still several places you can drive right on). The surf there is moderate and generally safe -actually you tend to find more surfers here than Daytona. Cocoa Beach has great surf and can be rough -maybe not the best for inexperienced ocean swimmers. It does have Ron Jon's which is worth the visit, however. I love the Gulf Coast beaches as others have mentioned, but it's not the ocean and offers virtually no surf. Check it out on the web, I'm sure they have a pretty good website. Also, I've heard pretty good things about a casual seafood restaurant in New Smyrna called Boston's Fish House(they have a website, not sure if I can post here). The only thing I ask is please don't tell too many people about New Smyrna -it's kind of our own little secret!
Depending on age of kids, I recommend going to a no cars beach. If you do go where cars park on the beach, set up your chairs and lay on the towels etc. only behind your car not on the side between cars. SUVs can't see bodies lying on towels and again this year a sunbather was run over.
If you have kids set up on the water side of the" road "walking your kids through the traffic and warning them that they cannot run back to the car for any reason without you.
We love New Smyrna. Beach is wide. Our favorite place to eat is The Breakers- a bar/restaurant. Friendly, good food but not a whole lot of tables because part of it fell in the ocean during the 2004 hurricanes. I think it is on Flagler where the surf shops are. I know there are lots of surfers at the inlet, but it is also the shark bite capitol of the world.
Cocoa Beach is also nice. There are parks south on A1A that are great for smaller kids. For example, Lori Wilson park also has a nature trail to walk through. I usually bring my nieces and nephews there because the bottom goes out shallower than in some areas where water is overhead and then maybe knee deep farther out etc.
Best spots to see surfers are mostly at the pier, 16th street, and down at Patrick Airforce Base at First light and Picnic tables(has a pavilion and yucky bathroom that doesn't really have a door so not real private.) You can rent boards at Ron Jons.
Go farther South to Melbourne area and you can see rock formations at low tide that are really cool, but sort of dangerous to go out in. Farther south to Indiatlantic and Melbourne Beach areas and you get less rocks, public beaches with pavillions and bathrooms, and you can usually find larger shells.
June is generally "Lake Atlantic " time on the East coast so not a lot of good waves. It also isn't as good a time to find bigger shells. Still a great time to be at the beach.
To see the best surfing Sebastian Inlet state park is the place to go if you don't go to the inlet in New Smyrna. There is a charge to get in the park. There is a little restaurant, a museum, the jetty, and a walk bridge that goes half way over the inlet where you can watch the small boats come in and out. Just a little south there is a treasure museum that is interesting, too.
