Did you enable your core to act as the DHCP server for your network after disabling it on the linksys?
Cabling should be something like:
modem > internet port on Linksys
Linksys switch port > any other switch in the house.
A set up like this would be used for a single ethernet on the core.
To use a dual ethernet on the core, you would need to connect the modem directly to the cores external port (eth0) and connect eth1 to the internal network. You also will need to set up the core as your gateway and maybe also PPPoE depending on your connection type.
You cannot connect both a external and internal network or subnet to the same switch unless you are using VLANs.