Get initials from full name
Other than the style (yeah, you mentioned it) I think that you can:

First a small change: replace this

    int firstSpace = fullName.indexOf(" ");
    firstName = fullName.substring(0, firstSpace);
    int lastSpace = fullName.lastIndexOf(" ");
    middleName = fullName.substring(firstSpace + 1 , lastSpace);
    lastName = fullName.substring(lastSpace + 1, fullName.length());


with this:

String[] parts = fullName.split(" ");


But the main change I would suggest is a design one: remove the need for instance variables, by making each function return the input for the next.

public static void start()
{
    greet();
    String fullName = askInput();
    String[] nameParts = seperateFullName(fullName);
    String[] initials = getInitials(nameParts);
    showResult(initials);
}


(and of course modifying each function to return the value they compute)

Why? 


It decouples each different part of the program, keeping them independent, specific, and therefore less prone to errors introduced somewhere else in the code. 
It helps also with testing, as you can easily test each function separately. 
It also helps with reusability as you could re-use each different part in other places.


Other than that, check the spelling in functions' names (For example: in seperateFullName separate is misspelt)