I'm just terrified that if I make one wrong move, a hair or a piece of dust could fall in there, and then the camera will be irreparably damaged and I'll never be able to use it again.

Fear not, young Padawan. Those of us who regularly use dSLRs can attest that simply having dust or hair fall into your dSLR body will NOT irreparably damage anything, and you can even get the dust/hair back out again with simple methods like a bulb blower, or more complex ones involving the arcane methods of sensor cleaning. But dust on the sensor is usually just something you deal with on a regular basis.

A few facts to ease your mind.

1. The sensor is not exposed when your camera body has no lens on it. A dSLR uses a mirror to take the light coming in from the lens and throw it up into the viewfinder. When you take a picture, the mirror flips up, to let the light go through to the sensor. But at all other times, that mirror is covering your sensor. You're really only exposing the guts of the camera when changing lenses, if you shoot mirrorless.

2. Your sensor is also covered by a glass filter. This filter serves several purposes, including filtering out ultraviolet and infrared light to keep them from throwing the colors off on your sensor (which is sensitive past the visible spectrum). But it also protects the sensor. When you "clean the sensor", you're not actually touching it, you're cleaning the glass filter over the sensor.

3. Your camera, if it's not just brand new to you, but also a relatively new model, probably has sensor shake technology that effectively clears most dust/hairs off the sensor before taking a shot. You also typically don't see debris on the sensor unless you're shooting stopped down into the f/8 and smaller range, because it's out of focus to the sensor otherwise.

4. Your brand new shiny shiny camera probably came with a body cap. You can use that to protect the guts of your camera until you're ready to actually mount the lens. There are also tricks and techniques you can pick up (i.e., turning off the camera, holding it face down so dust falls out, not in, changing the lens in the bag, etc.) to minimize crap getting in while you swap out lenses.

I live in Southern California, where the weather is typically dry, and wet dust doesn't weld itself onto sensors, and I change lenses with wild abandon all the time. I don't do any of the "best practices" methods--I just use a body cap if I think I'm going slow and fumbly while juggling lenses. And I mostly shoot mirrorless these days.

It's basically nothing to worry about, really. It just takes time and practice and putting behind you the big dent you put in your wallet when you bought the gear, and starting to get to know it as tools. Take a deep breath, screw that lens on, and get started clicking. The more times you do it, the less frightening it becomes.