I suspect that you won't necessarily save money by growing your own veg in the first year at least - there are a lot of start-up costs, since you have to buy your tools, etc. I've found that cheap tools tend to fall apart pretty quickly, and that old tools rescued from relatives' sheds need a lot of repair.
Growing from seeds is the cheapest way, but you also probably won't have a very high success rate with germinating seeds in the first year, because you're learning what types of veg do well in your soil. For example, at the moment I've got a mix of young veg in my veg patch from an autumn sowing - 100% germination success rate on broad beans (but some then got chocolate spot and had to be taken out), high success rate with lambs lettuce and winter salad, zero spring onions and out of a whole row of carrots (autumn sowing variety), 1 successfully germinated. So if you can find cheap vegetable plug plants (our local market stall does them), I recommend going for those in your first year, just to keep optimistic and make sure that you have success from something. There's nothing worse than putting all the effort in and having no plants grow.
It's pretty obvious, but the way to save money by growing your own is to work out what you buy already which is expensive, and then grow that. So raspberries are expensive to buy, but usually pretty easy to grow, and after a few years you can get a real raspberry thicket in your garden. Herbs are another winner, and little lettuce leaves.
If you count the cost of your time, I think growing your own probably actually costs more - but that doesn't stop me doing it