You probably know this by now: Statistics can be hard to find. Sometimes, you need to look in unconventional places to help you find the research you need.
Check government publications, even if your research topic does not appear related to the federal government. The U.S. government outputs a lot of useful, publicly-available data. For example, suppose you're researching the demographics of students who attend college. The National Center for Educational Statistics will be your best bet for that information.
Review general statistical databases. Some databases do not appear to have much focus or direction; they contain a wide-variety of statistics on topics that have little, if anything, to do with one another (e.g. Statista and OFFSTATS). On these websites, search for your subject to try and see if it has been collected.
Consider synonyms when searching. Suppose you need statistics on tourism in the United States. "Tourism" is clearly a useful keyword, but so would be "Vacations," "Vacationing," and "Leisure."
Use a search engine. Yes, a search engine like Google or Bing will help you when researching statistics. Internet search engines have very powerful algorithms that present you with the information you want right on the first page of results. Suppose you are researching smoking statistics. A Google search for 'smoking statistics' will yield basic statistical data from the CDC (Center for Disease Control). However, do not stop there. You will note that the statistical information is very basic, and your research may demand a more comprehensive statistical analysis. For that, you should review the National Center for Public Health Statistics, which an Internet source will likely footnote and reference.