Welcome to SE:Arduino.
Shields are usually designed so that the pins are still fully accessible. The shield has a set of male connectors on the underside and female connectors (same as your Arduino) on the top side. Just Google a pic of a shield and you will see what I mean.
Ethernet is not the only choice, there is Wi-Fi, Bluetooth, NFC, radio and probably a point to point light link. What ever communication method you select, you would need to use HTTP POST messages to your server, so your server will have to implement a REST or SOAP interface. (There are probably other ways, but I'm not a webbie so you will need to ask them on the correct SE forum)
Once you have created your message on the Arduino you send it to your shield, probably by a function call or over a serial interface. You'll get an OK back and all is good.
To find code that does this, just search, someone will have written and example, Adafruit and the Arduino.cc sites are good places to start.