Hello, and Welcome to GeeksToGo,
It sounds to me like you need to setup Port Forwarding to allow your xbox to access the internet through your router. Port Forwarding is Basically allowing Devices or Programs access to your internet while being able to bypass your routers internal firewall.
First, you will need to setup a Static IP for your xbox, so that way it uses the same IP everytime you connect. There is a Good Guide on this Here.
(Although Since you use xbox live you should be using the new dashboard NXE, it should be very similar to the steps listed just in the system settings folder on your Xbox Dashboard) You will need the Residential Gateway of your router, which should be 192.168.1.1 by default. Your IP address you enter in to the Xbox above the subnetMask should be similar but not the same as your residential gateway. I suggest something like 192.168.1.32 This way any wireless Computers using your network will not accidentally use the same IP in thier automatic connection.
Then you will need to move on to te next step, setting up the router for port forwarding. you will need the IP you entered into the Xbox right before Subnetmask. There is a good guide Here.
Although I do not know the default password for this router, it should be in the manual that came with the router. If not then u may need to call the company who produced it and ask them. If you need any further assistance please just ask.