Some herbs are easier to dry than others owing to their stronger leaves and oils. However, almost all herbs can be dried in one or another. Trial and experiment is the best way to find out which herbs dry best for you, with the expectation that some herbs will shrivel up and look like a brown mess when dried, while others will retain color and texture well.
The best time to cut herbs for drying is just before they flower. This is when the leaves have the most oil, which is what gives herbs aroma and flavor.
After harvesting them, rinse each branch in cold water and dry with towels or paper towels to remove all visible water. Wet herbs tend to mold which destroys the whole bunch.
I use a dehydrator because it makes things super simple. You just spread the herbs out on the racks, leaving space between them for the air to circulate, and only one leaf thick on each rack. Most Dehydrators have different setting for depending on what you are drying so you simply follow the directions on the dehydrator.
Some herbs such as basil, tarragon, lemon balm and mints have high moisture content and will mold if not dried quickly.
If you are drying your herbs in an oven you will want to dry them in a very cool oven (high temperatures will result in tasteless herbs). Basically, just turn the oven on to "warm" (140 to 200 F) (or 65 degrees C to 93 degrees C, gas mark 1) for 20 minutes, then turn it off and pop in the herbs.