The major component of this exercise is to learn how to interface with multiple sensors and interpret the given information. Like most projects, it can be as easy or complicated as you want. You can simply collect information from your Arduino's current whereabouts, or you can configure it to deliver that information to your computer wirelessly. If you're feeling really adventurous, you can even use the data to control your home thermostat! It's all up to you.
This is the most basic weather station. It measures temperature and humidity and outputs the information to an LCD. Even if you're planning to upgrade your station later, we'd recommend starting with this one to get the basics down.
This station measures barometric pressure as well as temperature and humidity for a bit more complexity. The instrutable also provides information on how to build a nice casing to store your station.
Use this instructable as a template for eventual thermostat control. It takes you as far as determining whether you're outside your comfortable temperature range - from there, you'll need to learn to interface with your own thermostat. This station also relies on a battery pack rather than external power connections.
If you'd like to send your data over the web, use this tutorial to get an idea of the protocols involved. This station measures only temperature and humidity, but once you have the web basics down, you can definitely expand its functionality.