Answer: Most vegetables will grow as well, if not better, in a raised bed as in a conventional garden plot. You can simply mound up the soil into long, flat-topped hills to create your raised beds. Choosing a building material can be tricky. Many people prefer not to use pressure-treated lumber for growing vegetable crops, fearing the chemicals used to treat the wood will leach out. Raised beds made from untreated pine will last a few years but will rot and need to be replaced. "Plastic lumber", made from recycled plastics is another option--perhaps not as attractive but certainly long-lasting. Concrete blocks or bricks can be stacked up to make a raised bed. You don't need mortar unless your bed will be higher than 3'.
A mixture consisting about 3/4 good topsoil, and the other quarter made up of aged compost, well-rotted manure and/or shredded leaves or grass clippings should work well to fill the bed. Depending on the soil type, you may want to add some lime to raise pH, and/or some bonemeal for phosphorus. Some root vegetables, such as carrots, require deep soil, so either make the beds at least 2' deep or build them up over existing garden soil.
An added bonus to having a raised bed is that you can erect a frame over the bed and drape it with clear plastic for an instant cold-frame to extend your gardening season.
Best wishes with your new garden!
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