For the user interface you are describing to be added to the input field, jQuery Datepicker springs to mind.
http://jqueryui.com/demos/datepicker/
You'll need to include the jQuery library as well as the jQuery UI library. You can use the 'build your download' link and documentation on the jQuery library page to get a fully themed ui download to suit your design needs.
You'll need to make your form post to a server-side script that will take the field and insert it into the DB. Be careful - mysql date time is in the following format - YYYY-MM-DD HH:mm:SS. You can use strtotime and date to format the input from jQuery. After sanitizing the input, you could do something simple like:
date('Y/m/d H:i:s', strtotime($_POST['date']) )
You can use this to make sure in your server script that the date/time is in the proper format before it goes in the database.