There are many reasons why you might be interested in cited reference searching, but for promotion and tenure purposes, you need to know how many times your work is cited by other authors--as a measure of your impact. The sources here will help you track and count citations, but there are some things you need to consider.
Do citation counts make sense for this research?
Citation counting isn't meaningful in and if itself, and it is important to remember that it doesn't capture all research equally. It is much harder to assess the impact of new research, books, software, and data sets (and other non-journal article work) with traditional tools even though they may be important to your discipline and have broader implications outside the academy.
Where does the citation count come from?
Different sources include different sets of publications. Some sources include citations from conference presentations, and some do not. These sets overlap. If you want to get a count of the total number of times something has been cited, you cannot just add the numbers together but have to look at the citations separately and remove duplicates.
Which citations count?
You've already learned that traditional citation tools aren't comprehensive. You also need to think about what they may include like self-citations and self-citation by co-authors and how that factors into your evaluation. Think about what citations should count, and how to capture them.
How can I avoid errors?
Common names (think John Smith) and name changes can make it difficult to track authors. You also need to account for miscitations--they happen more than you would think. Claiming publications on Google Scholar and ORCiD have the potential to alleviate some of these issues.