I want to save data from an Arduino to machine on the net.
Is the Arduino equipped with an Ethernet shield, or is it simply sending data to your PC and expecting the PC to talk to the "machine on the net"?
I guess I'd use processing to do that.
Not on a web server.
Would I use something to write to a MYSQL database?
Depends on the kind of data you want to save, and how you want to save it.
Next, to read that database and display that data on a web page, I'd need processing.js running on server?
Why? There are plenty of options, depending on the OS of the server.
Yes? You have a question about PHP?