Communication through internet?

Hello

I am looking for the basics for this type of device:
The electronic device would be linked to my home router (RJ45).
Everytime I press the button on the device, let's say it sends "hello" to a remote server over the internet.

Ideas?
Some steps to start?

Thanks!

Philippe

re:

I have seen my friend making similar type as you are making... He was using zilog microcontroller and it was really good. I wish I a have a record of the video and the circuit and his codes..

Found some elements here:
https://ladyada.net/make/eshield/

and here:
**broken link removed**

Everybody encouraged to contribute, all ideas welcome