You have to start by finding a spot for your pots that gets six to eight hours of sunlight a day and has access to water. I grow my veggies on the roof and in window boxes, but patios, doorways, sidewalks, and even roof eaves (for hanging baskets) can house a few containers. Most landlords are happy that tenants actually want to improve the property, but it's a good idea to get your container garden OK-ed before you start planting.
Related: Container Gardening
In general, shallow-rooted plants, such as lettuce, spinach, radishes, and most herbs, need only 6 to 8 inches of soil depth to grow well, while deeper-rooted plants, such as tomatoes and squash, need 12 inches of soil. Terra-cotta pots, wooden boxes, and even 5-gallon buckets make great containers. Just make sure your containers have drainage holes, are not translucent or opaque (sunlight will fry plants' roots), and are big enough to support the plants growing in them. Fill your containers with a well-draining potting mix (topsoil will compact in containers) that has some compost or an organic granulated fertilizer mixed in.
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Almost all vegetables grow well in containers, but choosing the right variety helps. Window Box Roma tomato plants, for instance, stay a size that's manageable for pots, and Tumbler tomato vines spill nicely out of hanging baskets. Beans, peas, and even squash can be grown up trellises set into a larger container. Try the compact Sunburst yellow scalloped squash and Spacemiser zucchini. Miniature White cucumbers have small vines and unusual white fruit. Carrots such as the heirloom Oxheart and the miniature Kinko grow to only 4 to 6 inches long. There are a ton of resources on container gardening, but the two that I turn to most are The Bountiful Container, by Rose Marie Nichols McGee and Maggie Stuckey; and Lasagna Gardening for Small Spaces, by Patricia Lanza. Both of these books have great information on soil, plant varieties, and plant combinations.
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If you want to actually sink your hands (and your plants) into the earth, try community gardening. You share a plot of land, as well as advice and friendship, with other urban gardeners. You happen to live in a city with a large and vibrant community garden culture, so there is most likely one in your neighborhood. For more information on community gardens, check out the American Community Garden Association (www.community garden.org, 877-275-2242).