The below information was taken from the archives of the news group, rec.gardens. The author (Victoria) is a Master Gardener in North Texas.
Subject: Hay bale Gardening-How to do it (long)
Author: Mark & Victoria
1. Where?
Answer. Bales may be placed anywhere in full sun for vegetables, in sun or partial shade, if specified, for flowers, as in any garden plan. Bales may be placed on any surface, e.g., patio cement, weed-choked or barren soil, on good soil intended to be gardened later, anywhere which permits normal drainage. If on a slope, avoid a floodway where bales could be washed out of place. However, a poorly drained area where rainwater seeps in slowly is all right, as the bale benefits from "wicking up" excess water.
B. A water source should be reachable with a hose.

2. Size and number of bales:
A. Use the standard small oblong block. Leave the bailing wires in place until the last crop is harvested; then the wires can go in the trash. If wires are removed prior to then, the bale will fall apart as it shrinks and softens late in the season.
B. Any number of bales may be used, arranged as desired for easy accessibility. They should be laid flat on the broad side, so that the top will accommodate two or three planting holes several inches in from any side surface. Bales can be stacked two high if a gardener needs to avoid even moderate bending.

3. What kind of hay?
Any type seems to work. (Straw CANNOT be used.) However, check with your supplier and avoid hay on which herbicides have been sprayed. This is especially important when children will be gardening.

4. When?
A. Place your hay TWO months before the desired date in your locale for transplanting bedding plants. This will be late January or very early February in North Central Texas. This allows time for the hay to soften somewhat and the heat of natural composting to subside.
B. ONE month before transplanting time prepare the planting holes.
5. Placement Process:
A. Do NOT use an herbicide to kill weeds in the area where you will place the bales. Instead, accumulate newspapers and cover the area where bales will be placed (also between row of bales in a large garden) with a continuous pad of newspaper. Unfold the papers to full size sheets and create pads 15 sheets thick. This barrier will control weed growth under and through the bales. Between the rows in a large garden cover the papers with leaves or other trimmings to keep paper from blowing around. As time goes on the paper will be wet through and stiffen; eventually it will become composted along with the old hay.
B. Do NOT spade or turn the soil before placing the bales. However, if there is weedy or grassy growth, scrape or scalp the area and rake away all loose material before placing the newspapers.
6. Preparing the planting holes.
One month before planting time decide how many plants you will grow in a bale. One bale supports 2 each of tomatoes, squash, melons, gourds; 2 to 3 each of plants requiting somewhat less space, e.g., eggplants, peppers, various cabbage family plants. Vining plants which have a spreading root system do especially well in hay bales. DO NOT plant root vegetables such as carrots or beets, nor plants needing regular soil such as peas, lettuce, radishes. Corn roots are shallow and cannot be properly supported in a hay bale. Hole preparation will be the same whether you will transplant a bedding plant or sow seed directly as with squash. Choose the recommended seed or bedding plant for your climate as you would with any garden. Hole preparation is critical to success.
Steps are;
(1) Dig a hole in the hay 4" in diameter and 6 to 8" deep. Stay well away from side surfaces of bale. The hay may be very stiff and compacted. A keyhole saw may be helpful in loosening the hay, which you then pull out with your hands. Or, you can loosen the hay a little at a time with a dandelion digger or similar too. Wear gloves, ouch!
(2) When the hole is large enough, fill it about half full with a mixture of one-half composted cow manure and one-half peat moss. (this can be mixed in a wheel barrow and wet to a mud, so that the peat is wet through.) Do all holes planned for a single bale at the same time.
(3) Use a hose and water this mixture into the surrounding hay.
(4) Partially fill holes again; hose the mix again. Repeat till the holes are filled and the surrounding hay impregnated with the soil mixture.
(5) Then, add a handful of red wiggler worms to each hole. This is not essential, but it is an asset in the composting process. (bait shops have worms)
(6) Planting;
After all danger of frost has passed transplant your plants into prepared holes or place seeds directly, as in any garden. Water liberally.
(7) Watering;
A drip system to provide continuous moisture in the meshwork of a hay bale is needed until fruit is strongly setting, approximately 6-8 weeks.
7.To make a drip system for EACH plant:
(1) Thoroughly rinse a plastic gallon milk, or similar jug.
(2) Punch a circle of 5 pepper-shaker size, needle-small holes in the bottom an inch or so in from the outer edge.
(3) In the center, punch a hole large enough to fit snugly and hold firmly in place a bamboo cooking skewer (available at the supermarket). About 2/3 of the pointed end of the skewer should extend out the bottom.
(4) Push the skewer into the hay securing the jug as close to the plant as possible without crushing it.
(5) This jug should be filled with water ONCE daily unless it rains enough to thoroughly water the bales.
(6) After 6-8 weeks these jugs may be removed, and watering done only as needed, as with a regular garden.
8.Trellising;
A. Any plants which will climb or be supported by trellising may be handled this way in hay bale gardening. One cost-free way to make a trellis is to cut bamboo stalks in the winter and let them dry while the bales are weathering. The twigs and leaves can be trimmed off and used to help hold down paper between rows, or go to the compost pile. The poles can be trimmed to the desired length when the trellis is built. In an area where bamboo grows, nearly everyone who has a stand of it will be delighted to have someone trim away part or all of last years growth.
9. Plans; May be as small as one bale or as large as 50 bales or more. Arrangements may be varied according to space or purpose. If you are planning a regular garden in, say, 3 years, a hay bale garden can migrate across your projected space and leave excellent compost in its wake. At the end of year three, turn under all of the composted hay to greatly enrich your regular proposed garden in its place. More modest hay bale gardens can be placed along a fence, on a patio, in a corner of a schoolyard. Just follow the same steps, and you can feel confident of an excellent harvest.

Notes: Other soil mixes will work. A good organic compost, alone, can be used to fill the holes. However, the foregoing is a proven system, having given many seasons of consistently luxurious harvest for the authors. After trying it once, you may wish to experiment. Chances are you will have great success with any annual plant with a root system which can send a network of tentacles throughout the hay to properly support the upper growth.