I'm completely ignorant on these matters, can I change my IP address? I have a few computers at home connected to a router and, from what I understand, all those computers appear on the internet as one IP address, right? Can I cahnge that IP address?
Your isp controls your IP address. Depending on your provider, unplugging your modem and plugging it back may give you a new external IP address. Why are you wondering about changing your IP address?
Generally restarting your modem will not give you a new IP address, it's controlled by your ISP and it depends on how long their leases are. I have Comcast, and I've had the same IP for several months even with several modem reboots.
The only way to change your external IP on demand is to contact your ISP.
If you have DSL then starting a new PPPoE session will sometimes give you a new lease. There's also changing the MAC on the WAN interface of your device to obtain a new lease.
There's little reason to, though, unless you're trying to circumvent some sort of ban or access restriction on a particular site or service.