If you're looking for really fresh vegetables and something off the beaten path to experience, you might want to try and find the hydroponics farm in Provo. We picked 6lbs of tomatoes, cucumbers, and basil for about $30 there. It was great to pick them at varying ripeness so they lasted our week trip on North. Bring a bag, or the guy there just gave us a shopping bag, and then you're allowed to roam free and pick what you'd like.
They don't get many visitors, and this is not the place to bring kids who may destroy a farmer's crops, but it's a very interesting place to understand how to grow vegetables on an island with poor soil. I personally grow hydroponic tomatoes in my laundry room in Chicago in the winter, so it was a treat for me to see a bigger operation.
I took a short video clip -
Here is where it is on Google maps, and it's not easy to find -
Head toward South Dock on South Dock Road, go all the way down South Dock Road until almost the end. You'll see a gas station called Caicos Oil. It's two roads before the gas station on your left (you'll probably end up at the gas station, then turn around and count the road right next to it, and one more back.) It's a dirt road, go down the road until it ends, turn right. Go down that road and make the first and only left. Drive up and make the first left through a gate, drive past what looks like a bunch of storage garages, and you'll see an opening in the fence at the end of them on your left.
It was great to be able to pick and eat fresh veggies in the dead of winter!