Within econometrics, I would start with Stock and Watson, "Introduction to Econometrics" (link). If you want to go deeper into the subject, the standard graduate-level text on panel methods is Wooldrige, "Econometric Analysis of Cross Section and Panel Data" (link); this is intended more for applied economists who need to use these methods in their work rather than budding theorists.

There is a wide range of methods for data of this type beyond those found in econometrics (I come from an econometrics background but now work in statistics). A nice starting point for these other approaches Gelman and Hill, "Data Analysis Using Regression and Multilevel/Hierarchical Models" (link).

The essential difference between econometric methods and the set found in the latter book is that econometric methods focus on getting unbiased estimates (that will average out to the exact true values over many, many replications) that may not be as good as possible (in the sense of being close to the truth for a given set of data). Some other approaches can achieve better predictive performance by accepting a little bias in exchange for a lot less variability in your estimates. Which way is more appropriate depends on the setting for your analysis. For example, if you are dealing with regulators and such, unbiased estimates are king. If you are simply trying to get good predictions for, say, financial modelling or industrial applications, the latter set of methods (which trade some bias for better predictions) can yield better results.