Suppose you have a parametric equation for a surface, for example:

x = 3*cos(s)+cos(t)*cos(s)

y = 3*sin(s)+cos(t)*sin(s)

z = sin(t)

The partial derivative of this wrt s is:

dsx = -3*sin(s)-cos(t)*sin(s)

dsy = 3*cos(s)+cos(t)*cos(s)

dsz = 0

The partial derivative of this wrt t is:

dtx = -sin(t)*cos(s)

dty = -sin(t)*sin(s)

dtz = cos(t)

Which of these is the tangent in the s direction, and which of these is the tangent in the t direction?

I would have thought the tangent in the s direction is the derivative wrt s above and that the tangent in the t direction is the derivative wrt t, but I seem to remember hearing the opposite.

You would have thought correctly. You should compute some simple example to convince yourself that you are correct.

Agree with Alvaro. You're right on about the tangents.