Q. Where do I find unemployment rates and labor force rates and trends on the Bureau of Labor Statistics website?
The Bureau of Labor Statistics is the best place to find this type of information. Browsing the site is sometimes better than searching it, and a good place to start is in the Subjects tab. In the Subjects tab, you will notice that there are areas for both Unemployment and Employment
Use the Unemployment subject area to find information related to unemployment statistics. Click on the subsection National Unemployment Rate and then pick Labor Force Characteristics and then Occupation and Industry. This will allow you to to identify unemployment rates in a certain industry or occupation. Be sure to look at the annual tables and not just the most recent charts and reports. Characteristics of the Unemployed, in the annual tables, for example, is very detailed. Note that what you think of as an "industry" may not match the BLS's categories. Have a look at their category page on industry names to pick the "right" one.
Use the Employment subject area to find information on labor force rates and trends. Two subcategories, Employment Projections and Employment By Occupation, will be of helpful.