Laptop -> Router -> Ethernet port
Ethernet shield -> Router (same as above) > Ethernet port
If I do ifconfig on the laptop, I get the inet address as 192.168.0.100. If I go online and do a "whats my IP", I get 18.250.7.131.
Which IP address should I assign to the ethernet shield? My intuition tells me make it something in the 192.168.0.XXX range, but I'm not sure (and it hasn't been working)
Indeed you router is performing something called Network Address Translation (NAT). What this does is that to the outside (internet) it appears all your home computers come from a single public address (18.250.7.131 in your case). The router also provides your home computer with private addresses presumably from the 192.168.0.1-255 range. Every time you try to reach an address that's on the outside the router takes care that it changes the source IP address of the packet to 18.250.7.131 and also redirects the returning packets to the right recipient. IP version 4 addresses (the ones in question) are running out rapidly and NAT is a mechanism to make them last a little longer.