Basically there are three parts to this, setting up the phant storage location, getting the data, and pushing it to the web. Aside from the hardware setup, this all should take about five minutes.
To set up the data storage location, visit data.sparkfun.com/streams/make and enter the appropriate information. Title and Description are required, and the Fields are the things you will be recording (e.g. humidity and temperature). You can change these later. For more information, check the documentation.
After you click Save, it is very important to make a note of the provided values since this is the only time you will get to see them. Copy them to a document and also use the "Send yourself a copy of your keys" option at the bottom of the page.
So now you have a free, Internet-accessible place to log your data. Of course if you'd like, you can even set up your own phant server.
Step two is developing a way to collect data. This could be something like a barometric pressure sensor connected to an ESP8266 module, a button connected to the GPIO pins of a Raspberry Pi, or data that are already available on the Internet. For this post, we'll use weather data from openweathermap.org.
Step three is to push the data. I like Python and I've recently discovered the excellent Requests HTTP library. If you're using a Chromebook or you don't want to bother setting up Python 2.7 and installing that library, you can use c9.io.
Assuming that you have Python 2.7 set up and the Requests library installed, here's some code that will log the current atmospheric pressure.
And there you have it, a few minutes to set up and a few lines of Python to start logging data to the web. Next I'd like to see this all done with a sensor or two connected directly to an ESP8266 module.