Good sources for free current weather news include Weather Underground, The Weather Channel, Intellicast, Yahoo! Weather and the mother of all current weather sites, Weather.GOV by the National Weather Service. You can find links to more weather-related Web sites through My Virtual Reference Desk.
Historical info: Weather Underground makes it easy to look up historical weather data from the past few years for free. Look up the current conditions for the place in question, scroll down to the "History & Almanac" box, choose your date, then click on "Go."
The Utah Climate Center also posts a free historical weather data.
For more breadth and depth, the National Climactic Data Center (NCDC) has an excellent Web site that gives historical weather conditions for specific days into the 19th Century in the U.S., back to 1994 for foreign locations and, for some locations, monthly summaries back to 1697. Much of the data is National Climactic Data Center. Much of the data is free, but there is a small charge for the some popular reports. For the complete weather data available on any day, go to the Most Requested section and chose the relevant report under "Local Climatological Data, U.S." (For data since 2005, I go to the "Most Requested A" box and choose "1. Quality Controlled"). You could also go to Find a Station, then look in the "DATA" section for the report you want (not as reliable). You can purchase a certified copy (to be mailed later) after you get an online report. An explanation of NCDC abbreviations is posted under the Document/Decode links.
You can call the National Climatic Data Center if you have questions using their web site (828-271-4800). You can also hire the NCDC to compile reports for you. Data is fairly cheap if you can wait 2 weeks, fairly expensive for a same-day service.
Alternatively, most newspapers have weather forecasts and "Yesterday's Weather," though often without much detail.
Marine/Ocean Weather: See Ocean and Weather Internet Sites and General Information by Jenifer Clark.
Serious Research: For more scientific weather information, check out the Web site of the National Oceanic and Atmospheric Administration, which is the parent agency of the National Climatic Data Center (NCDC). I go straight for the site map at www.noaa.gov/sitemap.html.