We are new to Tableau and have developed a simple dashboard using Desktop we want to share with a customer. Customer has the Reader version but wants drill-down capability. What is the best option for us to give them this capability?

Create an extract of your data and then create a packaged workbook (a TWBX file) that contains both your vizzes and your data. Reader can ONLY use a TWBX file, but it contains all the interactivity that you've built into the workbook.