This process is simplified by having Chat on:
From Gmail, go to (Gear), Settings, Chat tab, Chat on, Save.
On the left of your mailbox, you will see something like this.
At the top of the list is you! Click the + and type (or select) the name of the person with whom you wish to initiate a hangout. This will start a pop-up window for a text-style hangout. To start a video call, click on the picture of the video camera. This will send a notification to that person, who will need to accept your hangout.