Surprisingly, growing your own food DOES save money.
It's not surprising -- it just isn't worth it for most people. To do it well, you variously need land; upkeep time; knowledge (pests you don't need, creatures you do, plant nutrition, how to harvest without doing damage, control of wastage, fertilizer issues, varietal information, home-cooking skills, canning skills); seed sources; patience; storage, fencing to control animal forage, sometimes a permit...
Or you can just go to the supermarket, buy a bag of salad and a can of beans, come home and cook dinner. Or hit a restaurant.
It's pretty easy to see why most people choose to exchange the labor they do via the obvious proxy (money.). It really depends where you want to put your effort. The money you save -- whatever that is in a particular case -- has to be of at least the same value as your time, otherwise, you're working against yourself.
We have a tower garden here. It was a gift, so the initial cost (to us) was nothing. Even so, the costs for the nutrients and starters and the small amount of electricity the nutrient pump takes adds up to be non-trivial, and the amount of produce isn't fabulous overall, all things considered. The quality of what it produces is, though. Buying it... I wouldn't even think of it. It's expensive. It's also kind of pretty when it's all growing like a little vertical jungle, but that's pretty minor in the larger picture.