I'd like to know how I can wire the Arduino to some sort of switch that can handle the voltage (is that the right word to use here?) of a wall outlet.
That would be a [u]relay[/u]. A relay is an electrically-operated, electrically-isolated switch. For your safety and the safety of the Arduino, it's important that there is no direct electrical connection between the Arduino and the power line.
But... Most relays can't be directly 'driven' by the Arduino's 5V, low-current, output. Assuming you don't want to build your own driver, you can get a 'relay board' that has one or more relays and associated driver.
Or, an easier option is a solid state relay. There are lots of solid state relays that can be directly driven by the Arduino. [u]This "industrial" style[/u] solid state relay is easy to wire-up (with screw terminals).
...Just FYI - There are a variety of home automation systems/protocols. For example, you can replace a regular wall switch with a home-automation switch/dimmer that you can control manually like a normal switch, or remotely. [u]X-10[/u] has been around for something like 40 years, and is still popular. (I have a combination X-10 & Insteon system.) Generally, you wouldn't communicate directly with a switch or outlet, but you'd have a special "hub" that you can access via Wi-Fi. So, you could build an Arduino gizmo that communicates with your "home automation" via Wi-Fi.
My girlfriend was recently recovering from a broken leg and I set her up with a couple of [u]wireless remotes[/u], a couple of [u]relay switches[/u] and a [u]transceiver[/u], to control a couple of ceiling fans. My system has a lot more controlled "stuff" and a master timer. (Neither of us has Wi-Fi or any kind of cell-phone connection, but that could be added.)