I have a web-form where users can select a date from a calendar pop-up and a time from a dropdown. At the moment I am trying to store the date using a Date object.
Have a look at the Calendar class. It has all of the methods required to get different "date parts". Also, look at the SimpleDateFormat class in java to format the date in needed way.
Calendar - http://docs.oracle.com/javase/7/docs/api/java/util/Calendar.html
SimpleDateFormat - http://docs.oracle.com/javase/7/docs/api/java/text/SimpleDateFormat.html