If both computers have simultaneous internet access via the router, then they are "networked" and there is nothing wrong with your connections.
The "1394 Connection" is Firewire. It has nothing to do with your interent access and, although it can be used to network two computers, it is unusual to do it this way.
Ignore the "Internet Gateway" entry, which should show as "Connected", unless there is a problem.
Your actual *network adapters* with which you connect to the router are the "Local Area Connection" (for a "wired" connection) and the "Wireless Network Connection" (obviously for a "wireless" one). A computer can have one or more of either type installed.
As regards "File & Printer Sharing", make sure you have at least one folder "shared" in each computer, by right clicking and selecting the Sharing & Security option. Any firewall software in either computer needs to be configured to allow access to the networked computers, which may involve adding their IP addresses in a "trusted" section. An incorrectly set up firewall is the most common reason for computers not communicating in a network.
SETTING UP A NETWORK PRINTER:
1) Connect the printer to the "client" PC, as though it were to be used with that computer, and install it. This ensures that it has all the necessary software/drivers on board.
2) Reconnect it to the "host" PC and, in its Properties, set it as "shared". In the client's "My Network Places" (or Network Neighborhood), the printer should then appear as a shared resource of the "host" PC along with any shared folders. Double clicking the printer in the remote PC should automatically install it as a "network printer" in the "client". This means that the "client" now has the same printer installed twice - once as a "local" printer and again as a "network" printer (with a bar under the icon to indicate a network device). If you are always going to print via the network, you may want to set the "network printer" as the default.
Otherwise, have a look at this Microsoft article. It is an old article written for Windows 95, but the basic principle for installing the network printer should still be valid.