I have a linode in the Dallas data center. I was curious where it was actually located, so I used
http://whatismyipaddress.com
It said the IP address that my linode is on was actually in New Jersey. But when I do a traceroute, either to or from my linode, it goes through router1-dal.linode.com or router2-dal.linode.com. http://whatismyipaddress.com says that routers 1 and 2-dal.linode.com are both in Texas where I figured they'd be. So my question is, is there some kind of VPN that routers 1 and 2-dal.linode.com are connected to where the actual physical server that the linode is on is actually located in New Jersey? I'm guessing the reason why data centers exist in multiple places anyway, is so that the connection between you and the server which you're communicating with doesn't have to be so far away.

Geolocation by IP address is fraught with inaccuracy. Linode HQ is in New Jersey, which is why your lookup incorrectly gave that for your IP address.

As far as I know, connections between the data centers Linode uses go over the same public Internet as other traffic; there is no special "direct" network linking them.