I'm an amateur/hobbyist photographer and a novice in terms of dragonfly/damselfly photography. Nevertheless, I will share a few points from my experiences:
1) You'll want a medium telephoto lens with a small minimum focusing distance (MFD). I don't shoot enough where I'll buy specialized lenses, but if I were to do so, I'd get a Canon 300mm (f/4 or f/2.
lens. With my limited equipment, I mostly use my 400mm f/5.6, with one or more extension tubes (typically 36mm) to reduce it's very long MFD, but that is far from an optimal setup. I sometimes add a 1.4x extender which doesn't change the MFD and allows me to bring in some more distant shots (e.g. damselflies sitting on lilypads). Finally, I sometimes use the 70-200mm with or without extension tubes as well.
2) A borrowed a 100mm macro lens briefly and found that I wasn't able to use it as a true macro very often with most dragonflies since they are a usually too shy for closeups. However, I like imaging other types of insects as well, so I'm seriously looking at the Sigma 150mm macro. I've attached the best macro shot that I was able to get with it.
3) Shallow depth of field can make for nice images, but in some instances it can make it hard to identify the species. So, if identification is important to you, you'll want your settings to allow the entire insect to be in focus. Or, vary your settings if you have a cooperative subject.
4) Be patient. As others have said, if you scare one off but you remain still, it will return to its original spot.
Here are a few of my photos. I don't recall which ones I use the extension tube with:
Azure Bluet, 5DIII, 400mm f/5.6 lens
Halloween Pennant, 5DIII, 400mm f/5.6 lens
Halloween Pennant, 5DIII, 100mm macro (non-IS)http://bartolini.zenfolio.com/img/s6/v146/p425824408.jpg