Reddit mentions of A Stata Companion to Political Analysis

A good resource for you might be Pollock's Companion to Political Analysis. It's political science examples, but it goes through basic analysis (describing data, regression, t-tests, etc) and is very easy and clear to understand. I already knew stata before being introduced to the workbook but (nerd alert) I found it fun to work through the workbook. Yes, I did the workbook for fun (and because it is used in a class I am TA'ing and I did not want to be unprepared). The workbook is available on Amazon

In reading your original post, it seems that you may also be looking for original empirical research articles where the code is presented in the methods and the data is available so that you can run the same code and view the same results. That may be harder to find, because usually the process is described without the code (e.g., "The regression model was estimated using Stata 11.0 SE..."). For articles that use publicly available data sets, I'm sure you could read those articles, download the data they used, and replicate the results by looking up the code.

There are some articles that do provide the code that are intended to be instructive in nature. For example, there is a great article by Singer (1998) which presents how to use SAS to generate multilevel models. Those might be more helpful as you find particular methodologies you are interested in, but as far as learning the basics (how to use stata) the UCLA website/workbooks like Pollock might be more helpful for now.