What precautions should I take when working with the hardware inside of my computer?
You need to safeguard yourself from static electricity. The components in your system are very sensitive to static electricity and can be damaged by it very easily.
Precautions you can take
Purchase an anti-static wrist strap and clip it onto the metal frame of your computer case while working with the components inside.
Purchase an anti-static mat, so you can work inside of your PC on the mat safe from static.
If do not have the ability to get the wrist strap or mat you can still take other precautions, however they are not 100% guaranteed to work in every situation.
You should try to work inside of your PC while standing on a wood or tile floor as you have less chance of building up static electricity on these surfaces. Avoid working on your PC on a carpeted surface without the wrist strap or mat.