Before we start, just a simple example of a calculation in Mathematica. To execute the cell below, place the cursor in the cell below and press ⇧+⌤.

2 + 3

This is how commands are executed in Mathematica.  Place the cursor in the cell with the command and press ⇧+⌤.

How to plot graphs of two-variable functions in Mathematica?

The easiest way to do this is to use the command  Plot3D.

The first few examples

In the example below change the  blue  formula to plot a different function. 
If you want to plot the function over a different domain, then change cyan numbers. 
If the picture is too small or too large change the magenta number.

Plot3D[x^2 + y^2,{x,-2,2},{y,-2,2},BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

Plot3D[x^2 + y^2,{x,0,2},{y,0,2},BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

Plot3D[x^2 - y^2,{x,-2,2},{y,-2,2},BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

Plot3D[1 - (x^2 + y^2),{x,-2,2},{y,-2,2},BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

Plot3D[y^3 + x*y,{x,-4,2},{y,-2,2},BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

A few more options

In the last example it would be desirable to change the  PlotRange. This is done as follows

Plot3D[y^3 + x*y,{x,-4,2},{y,-2,2},PlotRange -> {-5, 5},BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

In the last example I don't like how Mathematica deals with the values of the function outside of the prescribed range.  To get the graph how I like it,  I change the option  ClipFill. At the same time I add more PlotPoints.   This is done as follows

Plot3D[y^3 + x*y,{x,-4,2},{y,-2,2},PlotRange -> {-5, 5},ClipFill -> None,PlotPoints -> 60,BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

Finally, I change the default ViewPoint. If you want to change the ViewPoint again, then do the following: (1) highlight the entire  ViewPoint->{-2.634, -1.919, 0.910},  click on the menu item  "3D ViewPoint Selector" in the   Input  menu on the menu bar  (or, simply press ⇧+Ctrl+V), (3) move the box to desirable position (4) click Paste.

Plot3D[y^3 + x*y,{x,-4,2},{y,-2,2},PlotRange -> {-5, 5},ClipFill -> None,PlotPoints -> 60,BoxRatios -> {1, 1, 1},ViewPoint -> {-2.547,  -1.856,  1.231},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

A problem with a domain

The function f(x, y)  =  (Sqrt[1  -  ((x^2) + (y^2))]) is defined for (x, y) such that (x^2) + (y^2) ≤ 1. Mathematica knows this, but it has a hard time dealing with it.  To get the visual impression of a sphere, I use BoxRatios->Automatic.  Ignore the warnings; Mathematica is just telling you that it can not plot nonreal complex numbers.

Plot3D[ 
      Sqrt[1 - (x^2 + y^2)] , {x, -1.1, 1.1}, {y, -1.1, 1.1}, 
      PlotRange -> {0, 1}, PlotPoints ->  120, 
      BoxRatios -> Automatic, ClipFill -> None, 
      ImageSize -> 300, 
      AxesLabel -> {"x", "y", "z"}]

Two more plots

Plot3D[x y Exp[-(x^2 + y^2)],{x,-3,3},{y,-3,3},PlotRange -> {- .4,  .4},PlotPoints -> 60,BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]

Plot3D[(x^3)-x-((y^3)-y),{x,- 1.5,1.5},{y,-0 1.5,1.5}, PlotRange -> {-2, 2}, PlotPoints -> 60,ClipFill -> None,BoxRatios -> {1, 1, 1},ImageSize->300,AxesLabel -> {"x", "y", "z"}]