The modem component is only used for connecting to the internet, and is usually provided by your ISP (cable or DSL generally). Your router will be used to be able to connect multiple computers (to one another, and potentially to share one internet connection).
With your new router, you will be able to connect your computer to the router without any additional requirements. However, if you want to connect to the internet, you will need some way of connecting the router to the internet (a modem).
Most ISPs will provide a modem with your service free of charge (when you have a subscription to high-speed internet access). Modems generally have both a USB and an Ethernet port on the back.
You will need to connect your modem to the WAN port on your router (using an ethernet cable). Then, go to the router's configuration page (I think it's 192.168.1.1 for LinkSys), and enter the appropriate information (for DSL, you'll have to configure PPPoE, for cable it's usually DHCP). You will most likely not be able to access the configuration page via wireless access until you have configured your router to be able to do so. I would recommend making all your changes by plugging in directly with an ethernet cable, then once it's all working, using your wireless.
If you're having trouble, give your tech support service a call and they should be able to help you out.
I would highly recommend using WPA2 encryption on your wireless network. If you don't use any protection, you can have other people connecting to the net through your account, or even accessing the information on your computer!
Also, be sure to change the administrator password on your router. If you don't, you run the risk of having someone else change your configurations or again getting access to your information.
