In Incorporated Newark on Main St outside the Newark Natural Foods running all summer long from 10-2
Newark Natural Foods Cooperative - The Newark Co-op | About the Market
They also have some music as well.
Also, Ramsey Farms in North Wilmington (right off of 92 and 202) has had fresh strawberry picking in the past (although the web site is still showing pumpkin picking)
http://www.ramseysfarm.com/index.asp
Milburn Orchards in Elkton is another local place that has a market
http://www.milburnorchards.com/
warning - I have bought stuff from here in the past and it barely made it home (melons collapse a few hours after being bought - might have been a bad choice on my part but it happened more than once) - also, since the store tends to be open air, fruit flies get in and you may bring them home with you
But they do have pick you own peaches, apples, raspberries, etc.
Also, if you like blackberries, look around your neighborhood...chances are that the woodline border to the common areas is filled with thorny vines...the berries will form as pink, then red, then dark purple - pick them then (be sure to wear long sleeves and gloves as those thorns HURT) - nothing like free fruit for 15 minutes of work (and I mean you can get a lot from a couple hundred feet of woodline)