Dinoluvsoup asks via Twitter:
This depends mostly on what kind of food you are dehydrating. You have essentially three options: air dehydration, heat dehydration, and cold dehydration.
One of the simplest is heat dehydration, because you probably have what you need at home: an oven, a baking sheet, a cooling rack, and something like aluminum foil or parchment paper. This is what I use for tomatoes. Start with the baking sheet, put the foil/paper on top of that, then the cooling rack on top of that. If your tomatoes are larger than cherry or small Romas, cut them to a fairly uniform size so that they will dry in roughly the same amount of time. Put in the oven at 250°F or so and let them dry for a few hours until they have the consistency you want. I usually leave my tomatoes a little juicy and refrigerate them, but you can keep them going until they resemble what you would buy in a bag in the grocery store for longer storage.
The idea with this, and all, dehydration is that you want space around the item for air to flow to give water every chance to escape. Incidentally, if it's a nice warm day, you could also just put the tomatoes on a cooling rack outside and sun dry them.You use the heat of the oven or a warm day in this case because tomatoes are very juicy, and it would take forever to do otherwise.
The next relatively easy case is air drying. Useful for herbs and the like. My favorite rig is Alton Brown's, which consists of a box fan, some bungie cords, and air conditioner filters (the big 10" x 20" type with the waves in them). You get four or five of those, put your herbs into the grooves, stack the filters, and secure them to the fan with the bungie cords. Turn on the fan and run for hours, drying your herbs and being the best air freshener ever.
Again, the key is space between the thing you're drying with ability for air to flow around. This is especially important if you are not using heat at all.
Freeze drying, well, that one's trickier. It's a really great method for quickly removing the water from food and ensuring that the texture of the food is maintained. Like the other methods, the idea is that you want the environment to be very cold with air flowing around the food. I haven't seen a method I like yet for doing this without specialized equipment, but none of the home dehydrators really do this either, so I presume you're okay with not freeze-drying.
There are systems that you can build for doing large-scale warm-air drying. The downside of the oven drying is that there's not that much air flow relative to air drying, and that slows things down and requires more heat and time to make up for it. So what you can do is build a warm, forced air system. There are lots of designs for these, but I think you get the idea of what you need: space between the food, easy air flow, and a source of warm air.
A fantastic site for plans for things like this is Instructables. Chances are, someone handy has wanted to do just the thing that you want to do, and has taken photographs and written up steps to show you how its done. In this case, we have things like the Simple Fruit Drier. But that is merely one example. Do a quick search for dehydrator, and you'll find many more things, some which may fit with parts you have lying around the home, or some that may just resonate better with what you want to do. There are also discussions, some in the individual Instructables and some on their own, for how to make designs better or where to find specialty parts or what really happens on step three because either it didn't make sense or it caused the hair dryer to fail before drying all of the apples or whatever.
So your options are many and varied, limited only by your handiness, needs/desires, amount of free space, and patience of the person or people with whom you share living space.