Even a small garden can make a dent in your grocery budget.
Additionally, some of the easiest foods to grow, herbs, are also the foods that you pay the most for per pound at the grocery store. Perennial herbs are even better because you can just plant them and essentially not worry about them, aside from ensuring their watering needs are met.
Sage, mint (isolate it or it will take over), chives, lovage (huge though, you need space for this one), and oregano are some nice perennials.
I highly recommend perennials when it comes to vegetables too. They can help carry you through a year when you don’t get a good garden put it. Some options are; French Sorrel (some people would call this an herb, but I use it like a vegetable, so…), asparagus, nettles, rhubarb and Jerusalem artichokes.
Container gardens offer a lot of options if you don’t have a yard, as do community garden plots.
Planting your own fruit trees and berries is also an economical option. For very little maintenance you get a huge payback, after a few years of patience while the plants mature.
For faster results try berries first.
Some of my favorite resources are:
Square Foot Gardening by Mel Bartholomew
Eliot Coleman’s books