Don't buy things. Garden gadgets are fun and helpful, but usually not really necessary. Buy a shovel, a rake if you can, a hose and sprinkler and/or a watering can. Try to buy them cheap at a yard sale, and you'll get by.
Compost. Fertilizers are expensive, and so are compost bins, but a pile of compost will work fine. If you're concerned about the unsightliness of it, dig a big hole to toss the compost in, and cover it with dirt whenever it gets nasty. Dig the finished compost out whenever you feel like it and add it to your beds.
Buy the right seeds. Some people suggest saving seeds from grocery store produce, but that is generally a false economy. The vegetables you buy are varieties developed for commercial growers, usually in a different climate than yours. Buying varieties developed for short growing seasons will probably give a much better yield. Seeds for open-pollinated varieties are usually less expensive, and seed can be saved for next year. Some hybrids are extremely productive, and the modest extra expense may be worthwhile. Look for varieties with comparably low "days to maturity" numbers and words like "extremely productive".
Plant high-value or high-calorie crops. If you want to save money, plant high-value, productive crops such as tomatoes. If you just need food to eat, plant high-calorie crops such as potatoes, beets, parsnips and squash, most of which will also store well for winter eating.
Make the most of the season. Start seeds indoors (the package will tell you when). Put them in a sunny window, or make a cheap grow light with a high-wattage, daylight compact fluorescent bulb in a desk lamp. Starting your own plants is less expensive than buying transplants, and gives you a greater choice of varieties. It also allows you to harvest earlier from plants that are usually direct-seeded, such as lettuce. With vegetables that will be harvested frequently, such as greens, be sure to keep planting throughout the season so there will still be something to harvest into the fall.
Follow this advice and pray for good weather, and you should be able to cut your grocery bills. Oh, and learn to bake bread.