Are oxygen and carbon dioxide elements?

An element is a pure substance, which is made up of only one kind of atoms. It cannot be further reduced to simpler substances by any physical or chemical methods. An atom is the smallest unit of an element.

Elements are classified into metals, non-metals, metalloids and noble gases.

So, Oxygen is an Element.

 A compound is a pure substance made up of two or more different elements combined chemically in a fixed proportion. Example of Atoms of different elements forming a compound.

Eg: one atom of carbon and two atoms of oxygen combine together to form Carbon Dioxide (compound). C + O2 -> CO2

A compound has unique properties that are distinct from the properties of its elemental constituents. 

So, Carbon Dioxide is a Compound.