I guess in this case, although I'm not a "sys admin" I will move the thread continuation to the Networking forum, but before I do that... Let's see if one post can answer your question. If it requires more dialogue, I'll move the thread.
If your router is setup for DHCP, it should be assigning IP address to all computers connected to it and that should enable you to share files. You can check that by running Windows Explorer and "drilling down" into the Network to see whatever shares there are available.
To see what the IP addresses of the two computers are, the easiest way (usually) is to open the router's configuration screens and you will usually find (somewhere therein) a table of currently assigned IP addresses. Otherwise, on the individual computers, if you Run CMD and enter IPCONFIG /ALL you should see a complete list of everything about that computer.
Don't hesitate if you need more info...