I consider myself a pretty thrifty shopper and I shop around for the things I need. For most of my produce I either shop at whole foods or a farmers market. Most of my grains I buy from the bulk bins and I purchase most of my dairy, egg and meat items only when on sale. I choose organic, cage free, antibiotic free, and ethically raised if it is available but if not then I go for the next best thing- minimally processed. I buy a lot of organic pantry items from target or whatever store has them on sale. I usually will not buy something unless I have a coupon or it is in sale. That way I can stay within budget and still eat foods that I find nourish my body. 

I know that Applebee's is a chain restaurant, and not the most expensive of chain restaurants, but I'd hardly call it "cheap". And "microwaved taste"? Not hardly. I realize Applebee's isn't a fine dining restaurant, but it's definitely not low quality either.
Now, to answer the original question. Most of the time we shop at a small local grocery store. I have no idea how fresh their produce is, and I honestly don't care. It tastes good, and gives me what I need. When we (rarely) decide to do more than a day or two's shopping at once, we do it at Walmart because it's the only chain grocery store nearby, and we want/need to save money where we can. I wouldn't mind doing the shopping at somewhere like Trader Joe's or Whole Foods, but there's not one within 60 miles of where we live.

If a family has a backyard, then growing some fruits and vegetables in it would save some of the costs and guarantee that they're truly organic. If it doesn't but a neighbour does, then maybe the whole neighbourhood could start a program distributing fresh fruits and vegetables for everyone in the area for a symbolic fee paid for the one who's letting them plant said products in their yard.

I've lived in a small town all my life, where we have two small grocery stores that don't have the freshest produce, but the closest big chains are at least 20 miles away.  We do have a farmers market nearby, but only in the summer.  So my mom started a vegetable patch a few years ago, and grows a ton of tomatoes and green beans and lettuce and squash and a bunch of other stuff that she cans or freezes, and we eat throughout the winter.  Same thing with local u-pick fruit farms.  Not always "fresh", but at least we know exactly what goes into growing them.

Currently I do not buy my own groceries, but I have before, like if I want to make something extra. I shop at Trader Joe's and farmer's markets, but often at my corner grocery store that is independently owned, and sometimes Whole Foods and Waldbaum's. When I eventually buy all my groceries I want to try for more fresh and local produce, TJ's and WF. Never WalMart. Low cost always comes with a price.

@Emma - Walmart is not that bad. The low price is not necessarily due to poor product, but the giant's sheer purchasing power. They get a far, far greater discount than any of their competitors, hence the lower prices. You DO realise that Walmart is the largest corporation on the planet, yes? This gives them greater power with discount purchasing via volume orders. No other store comes even close.

@flapper_femme_fatale@xanga - And we have Targets come up to Canada. Not looking forward to that.

I get all my veggies for freeeee from the organic farm I've worked at for 5 years! And in the winter, well....if they aren't easily stored vegetables (potatoes, onions, winter squash, etc), canned or frozen...well, I mostly just don't eat it! Sometimes I cough it up and buy some lettuce if I really miss salad. Woooot farm life!

Meijer. I like Kroger Marketplace sooo much better, but it tends to be more expensive.

I use a service that delivers a box of fruits and veggies to my door once a month. (it can be more frequent but once a month is what works for me. The fruit and veggies come from farms up north, the are very fresh and I have used them for awhile, it's an amazing service.

The one I use is Farm Fresh to You. I just wish they did eggs. I LOVE farm fresh eggs.

I live in a poorer area of Philadelphia, and thanks to the student population, they finally put in a decent nearby grocery store called the Fresh Grocer. It was definitely needed in the area, for both locals and students. The produce is supposed to be good quality, but I've found it to be the same quality and selection back home where my mom has Weis, Safeway, Food Lion, and at one time Superfresh (not to mention a non-super center Wal-mart that as a very small selection). I can't really be picky unless I feel like taking public transportation elsewhere.

Whole foods, and farmers market. I have a 20% discount from working at whole foods. So, of course there!