I do go to the county sites. I also like the site www.linkpendium.com
which has pages for each state and county. It basically is a list of links, some free, some fee sites. www.familysearch.org
has lots of information, such as census and vital records.
Local public libraries are often helpful, some have newspaper and/or obituary indexes. You can google the name of the city or county + public library to find them.