I agree with bigjon, Miami is not the place to stay in Florida. I do like the Keys, but it doesn't take long to see all there is to see. You'll like it if you just want to kick back on the beach with a few tropical drinks.
In Florida, I really like Cocoa Beach. Plenty to see in the area, but pretty laid back at the beach. Nice people too. If you have kids you have to go to Orlando.
My two favorite places to go for the beach in the eastern U.S. are Cape Cod and Myrtle Beach. Really really really nice people at both places and great beaches. Can't get lost in either place. Cape Cod is very historic and very laid back. If you go there, stay on the cape, but definitey visit Martha's Vineyard and Nantucket. I especially like MV. Myrtle Beach is families and very touristy, but lots to do, especially if you like golf, go carts, eating seafood, shopping, and taking long walks on the beach.
Virginia Beach is not that great a beach, but there's a lot of American/British history and a lot to do within an hour's drive of the place. Colonial Williamsburg, Jamestown, Yorktown, Busch Gardens, the Great Dismal Swamp Wildlife Refuge, Water Country U.S.A., and the Chesapeake Bay.
One more good beach area is the Outer Banks of North Carolina, but only if you want to get away from it all and like to fish and go diving around sunken ships. Kitty Hawk is there and so is Roanoke Island and the "Lost Colony" for a little U.S./British history too.