jQueryUI has a nice DatePicker with icon trigger (a calendar icon opens the calendar, similar behaviour to Outlook).
On the other hand, Bootstrap 3\'s Input Groups a
I prefer to use label tag with for attribute to select the input field.
The HTML Label Element () represents a caption for an item in a user interface. It can be associated with a control either by placing the control element inside the element, or by using the for attribute.
More details and examples: https://developer.mozilla.org/en-US/docs/Web/HTML/Element/label
The value of for attribute will be the ID of a labelable form-related element, e.g. input. In my understand it will focus to your input, since the input already triggers with the date-picker. So, the work is done.
JSFiddle: https://jsfiddle.net/om01kgk5/1/