Essentially You have 4 main parts to the basic rocket.

(1)  You have a nose cone.  This piece is the major piece that gives a rocket its form.  You can buy them pre-made out of plastic.  I prefer to make them myself with some dowel on a bench grinder.
Plastic Nose Cones | Wooden Nose Cones

(2) The next most important piece is the body which is simply any tube made out of either plastic or cardboard. Pictures Coming Soon.

(3) The next most important piece are the guidance fins.  They can be positioned on a rocket body at equal intervals apart beginning with 3. This means that three fins minimum should ever be used on a rocket.  I prefer to use three only because the more fins you add, the more stable the rocket flies but the more heavier it becomes.  It does take practice but a three fin design is very effective when it is done properly.   These can be made out of cardboard, but if you are planning on attaining higher altitudes you had better make them out of wood or plastic.  
Plastic Fins                     Wooden Fins

(4) Finally there is the engine. These, as can be seen below, come in variable sizes.  It all depends on the weight of your cargo and the altitude that you want to attain.  And yes.  That little brown tube is an engine capable of delivering a straw rocket up to 30-50ft.

Thus forms the home made model rocket...