Ask Mathematica what 3 + 5 is; it prints back 8. Mathematica adds the In and Out labels; you do not type them. You end each line with .
The ^ stands for "to the power of".
This asks Mathematica to work out the inverse of a 2 x 2 matrix.
This asks Mathematica to integrate a simple function.
This asks Mathematica to solve a quadratic equation. The result is a list of rules for x convenient for use in other calculations.
This creates a 2D plot of a simple function.
Here is a 3D plot.
You can access many of the calculator features of Mathematica just by pushing buttons in standard palettes. Click this hyperlink to check out the Basic Calculations palette.