So, will it be easier to either use an Ethernet shield or buy an Ethernet Arduino in order to communicate between it and the browser?
Easier that what? The web browser normally knows nothing about a serial port, so the ethernet shield, and Arduino-as-server, is the only way for a browser to communicate with an Arduino.
Is there a knack to serial communication between browser and board that I don't know about?
Since we have no idea what you do, or do not, know about serial communication between a browser and a board, how are we supposed to know?
Other, of course, than the fact that the browser doesn't communicate with the serial port.