Do not confuse defining an automaton and drawing an automaton.
The drawing is intended as a support for better intuition, and because
it can often be read more easily in a single look. That is what
drawing are usually for.
However, you are supposed to actually (be able to) give a formal
definition, with the mathematical notation you learned, including
specifying the transitions (not necessarily one by one).
In your example you could say, after specifying the transitions for
the accepted string: this is completed by all transitions on terminals
not yet considered in each state, each going to $q_{die}$, and all
transitions on $q_{die}$ going back to it. Even this can be said more
formally.
It is up to you, and to your instructor, to decide on the proper level
of formalism. A drawing is usually enough in simple cases, as it is
clear how to get the formal mathematical definition from it.
But for some more abstract problems, drawings are inconvenient, and
the mathematical notation is much more easily used. Up to you to decide.
Constructing the automaton, as drawing or as math definition, depends
on the problem at hand. There are many way, and your problem may often
be to find the right one. Much of your course will be to teach you
ways of building automata. These are only early and very simple
exercises.
One point worth remembering. In order not to clutter drawings or
descriptions, the state $q_{die}$ (or some equivalent states) is often
onitted, as well as all transitions leading only to it (the reader is
supposed to complete as I explained above). This is convenient, but
dangerous ... some constructions or reasonnings based on automata rely
on the fact that all transitions are taken into consideration ... and
you may forget those that do not show explicitly.
