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.
"Breakout is a prototyping tool for exploring the intersection of the web and the physical world. The popular Arduino platform and the Firmata protocol are leveraged to enable users to access physical input and output purely from javascript."