What I should be aware before I touch my Arduino?
Turning the power off and taking proper static precautions also applies to Arduinos.  Anorton's tip about not resting it on a conductive surface is also useful.  Some other things to keep in mind:


Double check the polarities of any connections you make.
Keep a consistent wiring color code.  Use red for power and black for ground.
Calculate the expected current through all components before you apply power.  Check the datasheet to make sure it's within safe limits.
Calculate the heat dissipation for higher power components like power supplies and motor drivers.
Connect and test one small part at a time, instead of in one big bang.
Make sure the parts you buy expect the same voltage, or perform the appropriate conversion.


For keeping people safe:


Know where the fire extinguisher is.
Don't put cords where people can trip on them.
Use a proper stand to hold your soldering iron, and keep pets and small children away.
If your project has a propeller or something physically dangerous, build failsafes and kill switches into the system.
Be careful to charge any batteries properly, and not let them overheat.
Be careful what you touch while troubleshooting.  Arduinos usually don't deal with very high voltages, but inductors and capacitors can build up higher charges than you expect, and hold it after power is removed.