Not sure it's possible to progressively drill down, but you can create filters so that you can directly select a state, city, or zip code from the main map to focus on. And yes, it's all done from a single worksheet. I believe the simplest way to do this is to use worksheet actions. Under the Worksheet menu, select Actions... In the dialog box, select Add Action > Filter...
The next dialog box looks like this:
Name the Action at the top. Select the source worksheet to apply the action to (the one with your map). Change the button to the right to Select. Select the target worksheet to be the same worksheet and select the Show all values radio button. Under Target Filters, select Selected Fields. Click the Add Filter button. The next dialog box looks like this:
Select the source and target fields (the fields at the appropriate level of granularity for your filter - the state one for the top level action, the city one for the second level action, and the zip for the third).
You will end up creating three actions. When you click on a state, Tableau will highlight the state, and when you click it again, Tableau will remove the highlight and show the whole map. Likewise, clicking on a city or a zip will do the same. Hope this helps.