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A note about conventions for spherical coordinates. The conventions that the physicists and mathematicians typically use for spherical coordinates are different. Physicists tend to use theta for the angle down from the $z$-axis. Mathematicians tend to use $\phi$. Warn your students explicitly about this difference, it is a major source of confusion. (For more information, see: Tevian Dray and Corinne A. Manogue, Spherical Coordinates, College Mathematics Journal 34, 168-169 (2003).) Different conventions for the name of the radial direction in cylindrical coordinates cause less confusion for student.

A note about unit vectors. It is tempting, and even pedagogically useful, to define a basis vectors as the vector pointing in a particular coordinate direction that has unit “length”, e.g. $\hat x$ is the vector of unit length that points in the direction in which $x$ is increasing. However, it is more mathematically consistent to recognize that any unit vector is dimensionless; it has pure direction. You obtain a unit vector $\hat v$ by dividing the vector $\Vec {v }$ by its magnitude $v$. The dimensions in the vector $\Vec {v }$ and its magnitude cancel. In this way, it is possible to use the same unit vector $\hat x$ to represent several different physical quantities (e.g. velocity and acceleration) in the same problem. The dimensions are carried by the numerical coefficient that multiplies the basis vector. Confusion can arise, however, when we the numerical coefficient in a particular context is 1 and therefore unwritten. In practice, the distinction between $\hat x$ and 1 m/s $\hat x$ never causes any problems, but the kinesthetic activity described above can sometimes prompt and alert student to raise this question.

A note about ordering You may want to warm your students up with doing Cartesian basis vectors, so that they have a clear idea of where the origin of coordinates. In spherical coordinates, $\hat \theta$ is a tricky example because it generally points downward, and so I wouldn't recommend doing that one first.