Although I don't use LIVE anymore, I was a pro at connecting and networking. So I thought I should put this guide up.
Ethernet: Ethernet cables allow you to connect to a router through a cable. You can also use this to use a computer as a wireless router. So, let's first do a router connection:
Make sure your xbox is close to your router, and an ethernet cable is handy. My ethernet cable came with my xbox, so it might've came with yours. On the back of your xbox should be a slot your cable can go into. Put any end of the cable in the slot, and look for the same slot on the back of your router. Put the other end of the cable into that slot, and connect. It should get to your network, internet, and LIVE.
Use a computer as a router: It's not to hard, but it requires alot on your part with the computer. So you need an ethernet cable, like above, and your computer near your xbox. Using administrater's permission, switch your network to share, so other people can use the network. Then make sure you're connected to the internet, and connect the ethernet cable between the xbox and computer. Connect. If your network is shared, it should connect there. If you are currently connected to the internet on the computer, it should connect there, and have no troubles connecting to LIVE.
Wireless: So wireless xbox adapters can be bought for around 99-200 dollars, no problem. So make sure your house has internet on, and the room you want to connect from can recive the information. Scan for a connection, and click on the connection you use (if multiple, the strongest signal is probably yours) and it'll ask you for your WIP, WPA, or passphrase. Look on your wireless router for this, usually listed as "passphrase," or you can call your internet provider. Fill that in, and your xbox will have access to your connection. Then connect. Flawlessly connected to the internet, it should get you onto LIVE.
Use an adapter for a computer: On some 360s, you can use a computer's adapter to connect. First you need to configure it to the console, if the device isn't recognized, you cannot do this. After it's configured, do the steps for wireless, and you should connect.
Please leave feedback on whether this helped or not.