How to convert serial input to Ethernet via an Arduino Uno and Ethernet shield?
This should be achievable with the UNO and the Ethernet shield. For extra serial comms ports the software serial library works well, but you shouldn't need it to begin with as the arduino communicates with the Ethernet shield via spi. The mega has hardware uarts and more memory, which is useful as memory can run out quite quickly as your code grows.

Without spoiling the fun of working it out, try setting up the arduino as a server with a fixed ip. Then use software on the pc such as hyper terminal or putty to connect to that IP address. The arduino Ethernet library page has good code examples. E.g. http://arduino.cc/en/Reference/ServerConstructor