A good weather briefing starts with developing an awareness of the overall "big picture" before attempting to get a detailed weather briefing. At many locations, you can learn about the big picture by listening to the Transcribed Weather Broadcast (TWEB), Telephone Information Briefing System (TIBS), Television Aviation Weather, National Oceanic and Atmospheric Administration (NOAA) Weather Radio, Television and Radio Weather Broadcasts, and newspaper weather maps. But the Internet is probably the most useful tool these days to accomplish this.
Pilots are urged to use caution, however, when choosing from any of the many sources of weather information available. Many of these sources may not meet NWS/FAA quality control standards and should not be relied upon for making critical preflight decisions. Aeronautical Information Manual (AIM) paragraph 7-1-3 describes this issue in detail. There is also an article on this subject entitled, "Tips for Recognizing Reliable Weather Sources" on pages 5 and 6 of the following publication on the Aviation Weather Center web site: The Front Nov 2005 (PDF).
One good source of aviation weather that does meet these standards is the Aviation Weather Center web site. The advantage of being able to see text and graphic information visually and even print the information should be obvious, however, it does not replace the call to Flight Service. Be sure to call a Flight Service Station for the latest weather and NOTAM information. The importance of NOTAM information cannot be overemphasized in this day and age where a missed NOTAM can literally have fatal consequences. For information on types of weather briefings and what information to give the briefer, go to: Basic Pilot Briefings.