Answer: You can grow just about any vegetable--just grow what you and your family like best. One caution: be wary of starting too big--and getting overwhelmed and discouraged. Remember you'll be building on your garden for many years to come!
There's an excellent book available for beginning gardeners by Pat Stone, titled "Real Gardeners' True Confessions" (Storey Publications, Pownal VT). It's a how-to book that includes confessions from well-known and now-expert gardeners - all the mistakes they've made while learning to garden. You never stop learning, either! Another great beginners book is "Gardening for Dummies" by the Editors of the National Gardening Association (read more about it at http://www.dummies.com). It's a great and simple guide to the mystique of gardening. Specific to your situation, though, I recommend a soil test. You can either purchase your own from at a gardening center or mail order (Burpee's Seed sells one - check out their website at http://www.burpee.com), or have it tested by your local Cooperative Extension Service. The test results tell what nutrients are lacking in the soil, and what you need to add. In general, you can always use more compost. It contributes to the organic matter content of soil, and organic matter is crucial to keeping plants healthy, since it contains substances that make soil nutrients available to plants. It also helps keep harmful soil diseases and insects in check by maintaining a diverse and balanced population of microorganisms. So pile it on! And build a compost bin or pile of your own!
Another general caution is to not work on the soil while it is wet. You may think it makes sense to get out in the garden as early as possible, but if the soil is wet, you'll only set yourself back. Wet soil compacts easily, and it's hard to "fix" once the damage has been done. When you do work the soil, till or dig it to a depth of 12-24" deep. If there is literally no space between garden plots, then outline yours with a frame of planks set on the ground, or a band of mulch (newspaper covered with straw) to define boundaries clearly.
Since you're starting your garden mid-season, stick with veggies that mature in short timeframes such as lettuce, radish, spinach, etc. (Avoid long-season crops like corn and squash.)
Hope the above gets you started on a wonderfully rewarding new hobby!
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