So Google just released a new version of the Google+ app that has Hangout support. But how do I start a Hangout on my mobile phone?
It's a bummer but currently you can't. You can only join a hangout from the phone. From the Google+ help:
On Hangouts for mobile, you have the ability to:
You can't currently:
(The January 2012 update of the Google+ app changed this, however.)
That is actually not true. As of yesterday you can. Here are the instructions.
Make sure you have updated to the newest version of the G+ app. You should have version 18.104.22.168969 that came out yesterday.
From there you need to go to messenger and find the person you want to hangout with and send them a message. Once you send them a message you will see a hangout icon appear in the top right corner. See here for more information.