Arduino Uno with Arduino Ethernet Shield

If I connect my Arduino Uno direct to my computer I can 'listen' on the USB/serial port to data coming from the Arduino. I'm a little bit confused as to handle the same sending of data from the Arduino over/with an Arduino Ethernet Shield (or Arduino Wireless or Arduino GSM shield). I probably cannot 'listen' to a 'serial port' in this case? Is it correct that in this case I have to send the data to a server's (e.g. my notebook, my server) IP Address (e.g. a REST service, a message broker)?

Thanks.

Yes, but you could create a webserver with the Ethernet Shield and show messages on that page. Or a special status page with all kind of information about the Arduino.
With the Ethernet Shield, you are still able to 'listen' to the serial port, if you want to.
You can also write log messages to a log file on the SD card on the Ethernet Shield.
And you can send for example udp messages to another server, or send data to ThingSpeak.com They have graphs, but also a text frame.

The Ethernet Shield can be used to do all that at the same time, but the Arduino Uno will run out of memory quickly. Most of us combine an Arduino Mega 2560 board with an Ethernet Shield.

Thanks for the answer/info.
One extra question concerning 'listening' via the Ethernet Shield, or WIFI Shield: I don't understand how can I listen to the serial port when only having the Ethernet shield or WIFI shield but no direct (USB) connection between my Arduino board and my notebook?

Thanks.

Suppose you have a Arduino Uno, and attach an Ethernet shield to it.

The USB connector of the Arduino Uno can be used for the (virtual) serial com port, whether the Ethernet shield is connected or not.

A 'shield' is a name use by Arduino for a 'plug-in' board or a 'add-on' board. A 'shield' can not run on its own.

When only the ethernet is connected, you have to use something else. Using UDP would be my first choice, but that depends on what you have and how you would like to receive the data.

I think it might even be possible, to run a ftp server on your computer and have the Arduino send data via ftp.
Perhaps a miniature ftp server like: Baby FTP Server
Arduino FTP: Arduino Playground - FTP

Ok thanks.
I know a shield is an extra "plug-in" on your Arduino board. I was just wondering how I could listen on the serial port via an Ethernet shield (plugged on the Arduino) when there is no USB cable connected from the Arduino to my MacBook Pro. But as you mentioned and I thought (in my first message) I can call a REST service, send output message to a message broker, using UDP, ... So my interpretation of your previous answer was just a misunderstanding.

Thanks.

Okay, sorry for the confusion.
The 'serial port' is the 'serial port' (via usb plug). There is no serial port via the ethernet cable.

An alternative for the serial port can be bluetooth.