The following picture shows the common network topology of home networking:
There you can see some of the devices that I talked about. If you use cable internet then you need a cable modem which mostly be provided by the ISP.
You don't actually need a router if you just one to use 1 computer for internet. But if you want to share the internet connection, you'll need additional router for connecting the computers.
But if you use ADSL, you can use ADSL router, there are plenty cheap ADSL router out there. You don't need the modem.
After modem and router, you'll need switch which provide a lot of ports that you can use to connect your computers. The good news is that routers these days come with built-in switch ports, so you don't need to buy additional switch. Mostly the routers have 4 built-in switch ports.
If you want to connect using wireless, you'll need Access Points. Here you can connect your laptops, hand held devices that support wireless connection, and even your gaming console like PSP or XBOX360.
Another good news is that there are routers that also have a built-in Access Point and built-in switch ports - hah - you only need 1 device for your network.
To set the devices are easy work, some vendors also provide installation cd so you can follow all the instruction in the cd. But I promise you it is very easy to make your network up and running.