If you are looking at this and wondering why in the world you would want to grow vegetables vertically, you probably have a great big garden with room to spare. For gardens with less space, like apartment balconies or miniscule yards, going vertical makes sense. Small gardens use vertical space in the same way cities use skyscrapers---to reduce the footprint without reducing capacity. Forget all the rules about how vegetables are "properly" grown and take a chance on something a little bit different---you may find vertical gardening is addictive.
Use saplings or bamboo stakes to erect simple, rustic tee-pees by tying three to five of them together near the top. Spread out the legs of the tee-pee and interlace the legs with twine to provide additional supports for growing plants. Plant beans (or other vegetables) at the base of each leg, training or tying vines and branches to the supports as they grow.
Use south-facing wooden fences as support for sun loving plants. Screw small eye-bolts or cup hooks to the surface for use as attachment points for climbing plants. Or you can use the hooks--screwed near the top of a solid fence--to hang a piece of wooden trellis or scrap bit of wire fencing to train plants on.
Grow plants with a trailing habit---like squashes, cucumbers, melons and indeterminate tomatoes---upside down in planting bags raised high above the ground.
Run a rope or cable between two sturdy points, such as trees or buildings, on either side of the garden. Hang nets or old net-style hammocks from rope to ground (anchor the bottom of net with "U" shaped wire pins or stakes pushed into the soil) and train peas or beans up them.
Place several tall stakes (bamboo, saplings, pipe or rods) in the ground vertically every few feet. Cross lash others perpendicular to those at intervals to make an openwork fence. Train plants upward as they grow by tying them in place with soft cloth ties.
Create open "box" frames by pounding four posts or stakes into the soil in a square arrangement. Run twine or wire round all four stakes to support a single large vegetable (squash, melon, tomato, etc.) planted in the center.
Grow delicate or hard to pick plants, like strawberries, in hanging pots--fruits are easier to pick and stay cleaner when cascading from hanging pots than when sprawled on the ground.