The best way to visualize them would be to create a 3D rendering of the coordinate planes where a user can modify the bounds of each axis. However, that can be difficult to do on a calculator. If you are trying to understand the coordinate systems, this image may help:

In Cartesian, or Rectangular coordinates, points are defined in terms of <x,y,z> where the bounds are (-∞, +∞) for all three axes. In Cylindrical, points are defined in terms of <r,θ,z> where the bounds are [0, +∞), [0, 2π], and (-∞, +∞) respectfully. In cylindrical r defines the distance to the point from the origin along the rθ plane (z=0) and θ defines the angle of revolution around the Z axis. In cylindrical, θ starts on the axis and increases as you move counter-clockwise. Spherical however, is a bit different. In spherical, r is defined as the distance from the origin to the point. θ is defined nearly the same as in cylindrical, as the angle of rotation about the vertical axis (since there is no "z" axis). Φ, however, is now defined as the angle of rotation about the axis perpendicular to the axis of θ. Φ starts in the plane of θ and increases as you move counter-clockwise.

While it is possible to convert between coordinate systems, it can sometimes create messy equations. Points will almost always translate directly, but equations can become much more complex when moving between coordinate systems. For example, the line Y=X, Z=0 in rectangular is the same as θ=π/4, θ=5π/4, Φ=0 in spherical. The equation for a circle radius 2 centered at the origin in rectangular is $x^2+y^2=4$, where as in cylindrical, that same equation becomes $r=2$