Testing 1, 2, 3

Google Play has a built-in ability to do beta testing, without having to make the app available to the public.  From the developer dashboard, you create the app, then select "APK" from the menu at the side.  The screen that comes up has tabs for production, beta, and alpha.  "Production" means available to the public, while the test options will only be available to a selected group of people.  To do beta testing, select the "beta" tab, and follow the instructions to upload the APK.

But there are a few ambiguities that can cause problems:

Google Groups

This could cause trouble, since you may not know what they are.  If you're on Google+, you may mistakenly think they are referring to the "Circles" concept.  But no, "Groups" are another concept, and they are just glorified mailing lists.  That's good, because it means that your testers don't have to join Google+, or any Google services, you just need their e-mail addresses.

So you can create a Group, and populate it with the e-mail addresses of anyone you want to test your app.  You then have to connect the Group with your app: Still in the "APK" section, and on the "Beta" tab, click "manage list of testers".  This brings up a dialog, which asks you to "Add Google Groups or Google+ Communities".  That's misleading, because entering the name of your Group won't work; you need to add the group's e-mail address.  You can find that in the Group's "Settings" screen.  It'll be something@googlegroups.com.

Production vs. Published

"Production" means available to the public.  "Published" means made available to someone.  That is, when you upload your APK, it is not yet "Published", and your testers cannot yet get it. 

Worse, it's not immediately obvious how you Publish it.  However, if you're in the Developer Console, and you have selected your App, in the upper right, you may see a button that asks, "Why can't I publish?"  Clicking on it will tell you the remaining things you have to take care of.  Surprisingly, you have to add a variety of items relating to the App's marketplace appearance, even if you are still in Beta.  These include screenshots, and a description, as well as assurances that you've met Android's guidelines.  Once you've handled all these issues, the button becomes a "Publish" button.  Pressing it will make your App "Published," and thus available for testing.  Now if you press "Manage list of testers," the dialog will give you a URL that you can e-mail to your testers to get the App.  Surprisingly, you have to cut-and-paste this link, rather than it being automatically sent out.