A quadrilateral is any four sided figure.  Do not assume any additional properties for a quadrilateral unless you are given additional information.
 

A trapezoid has ONLY ONE set of parallel sides.  When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are not parallel.

Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.

Notice how the properties of a parallelogram come in sets of twos:  two properties about the sides; two properties about the angles; two properties about the diagonals.  Use this fact to help you remember the properties.

If you know the properties of a parallelogram, you only need to add 2 additional properties to describe a rectangle.

A rhombus is a slanted square.  It has all of the properties of a parallelogram plus three additional properties.

The square is the most specific member of the quadrilateral family.  It has the largest number of properties.