A few times I've received this question, usually from people whose work intersects with economics, yet without those people needing to produce econometric studies themselves. "How can I better understand the empirical papers I am reading?" I have a few suggestions:
1. Attend some (empirical) economics seminars first, to get a sense of what you need to learn and how discourse proceeds and what sort of points end up being contested. Subsequent class learning will be more focused and productive.
2. Often a good hands-on undergraduate class is more useful for these purposes than a graduate class. The latter might have too much econometric theory and theorem-proving.
3. The quality of an econometrics class (for these purposes, putting aside frontier work) is not well correlated with the quality of the college or university it is being taught at. The quality of the class is instructor-specific.
4. The quality of econometrics in the profession has continued to rise. That is good news, but for the purposes of this discussion there is a downside, namely that mistakes are much less transparent. For an increasing number of papers, it is hard to judge the final quality of the work without spending a lot of time on it. Whether or not a paper can be replicated is a more important question, given that more researchers are operating with frontier-level techniques.
Do you have other suggestions?
Addendum: Here is Mark Thoma's course, entirely on-line and free.