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One year we found ourselves in an apartment, which had a large porch at ground level. We purchased cheap plastic pots from the local discount store, and created a vegetable garden on the porch. We grew tomatoes, squash, and potatoes. It was definitely a learning experience.
Many items do not do well in pots, unless you fertilize them well. Water can also be an issue - you'll need to sometimes water twice a day to keep pots moist through summer days. And those moisture retention crystals DON'T work - in fact we found that pots that had them in them seemed to dry out FASTER than those without.
Lettuce does well in pretty much any pot that gives it enough room to send down a taproot. If you don't have room for the root to grow, the plant will never get very big.
Squash and tomatoes can be planted three to a large pot, and allowed to spill over the sides - good fertilization is critical to both of these, in pots.
Potatoes are ok in any large pot that has room to hill up the dirt in layers as the plant grows. Some people grow them in garbage cans, putting the potato start on top of about 1 foot of soil in the bottom, and adding another four inches over the top. Then as the vine grows, they add more layers of soil. The potato will grow spuds off the entire long buried stem this way. When the top dies down in the fall, just dump out the can to harvest.
Good drainage is essential in pot gardens, but you need to make sure that you also use soil that will retain the water enough. Many potting soils dry to a water repellent brick within days. If you can mix potting soil with compost and manure, you'll get better results than with just potting soil alone. Tomatoes and squash love being potted in a shovelful of pure compost.
One advantage of pots is that you can move them indoors at the end of the season, and get an extended crop. You'll have problems setting blossoms after they move indoors, but fruits that are already forming will continue to mature just fine.