Assateague is awesome. But... I actually wouldn't recommend it as an intro to camping for little kids, because the mosquitoes are MONSTERS. I know my kids would be whining the whole time.
I've only camped on the national park side, and it has been a while. But, you are camping basically between dunes behind a fence on each side, so no ponies will be showing up at your campsite. They do show up on the beach looking quite picturesque. And it is awesome to roll out of your sleeping bag and go swimming. But the mosquitoes. They will find the one square inch that is not covered in DEET and bite it all up. It is supposedly better on the ocean side than the bay side, which I believe. And if you go before it gets too hot, then they're not as bad. But ocean swimming is kind of the fun of the place, so...
Lake Anna is a cooling pond for a nuclear reactor. It's pretty, but, IDK, that kind of weirds me out.
You might try this place:
http://www.covecampground.com/ There is a nice lake for swimming/fishing. You may encounter some interesting characters, but just make sure you tell the guys at the front office you are with your family & they will steer you clear of the more "party" oriented campgrounds on their property. I rented a cabin there once and it was very quiet and nice.