How can I find out how often my publications have been cited?

Google scholar works pretty well, though with all of these things there are edges cases that cause problems (non unique names, random poster presentations showing up counting as citations, etc.)

I find that ResearchGate tends to be pretty good. People can comment, annotate, and more generally fix up the metadata issues on this site in a social media style, and so it tends to be pretty accurate. However, since it's still quite new most papers are missing from its databases (though it still tends to count citations correctly?).

One of the official databases is Web of Science. It only includes citations of the published (not preprint) paper.

In mathematics there is the extremely reliable database MathSciNet. The drawbacks are that it's not freely available and it takes some years for papers and citations to appear there correctly. However, it really is accurate (they have staff to handle difficult cases with similar author names, authors changing names and stuff).