This should be reasonably simple to achieve using javascript (i.e. it will run on the user's PC without a round-trip back to the server).
Put all the controls on the form, but for the ones that you do not wish to display initially use the CSS attribute {display:none} and they won't show.
Then use the onclick() event of the control which will determine whether other controls are made visible to show / hide controls as you wish.
There is an example of something similar to this (not within Joomla, but there is no reason (that I can think of) why this won't also work fine within a Joomla module / component) here: http://www.java2s.com/Code/JavaScript/Development/Makebuttoncontrolvisibleorinvisible.htm