I attended basic and advanced weather spotter training offered through the National Weather Service a couple of years ago. It was held at Cisco and was a very interesting day for a small town person like me!
This is the link for classes that are currently being offered:
Other WRAL Top Stories
http://www.erh.noaa.gov/rah/skywarn/
The advanced sessions can be hard to find (I am not sure they even offer them any more). There is a link on this site to contact Jeff Orrock about setting up a weather spotter training session in your area if one is not being offered. At any rate, there is loads of information on this site, including a section called online weather school.
If you find an advanced session, I would recommend doing a little research beforehand to make sure you are familiar with some weather terms. Advanced training is NOT necessary to become a weather spotter.
Happy spotting!