Would I need to do anything in my settings of the wireless SMC Barricade to use
it as a wired router from my cable modem (since it is slower than my
new LinkSys Wireless Router - my new access point). What settings
would I need to set to "tell it not to broadcast" anymore?
Assuming all you have to do is turn off wireless -- look for that setting in the wireless router's user interface -- usually accessible via internet explorer at an IP address like 192.168.0.1. or 192,168.1.1 etc. Once the router login box appears User name will probably be admin and Password will probably be password or admin (unless you've changed it from defaults).
If in doubts have a look at any CD which came with the router or download the user manual from Linksys.