How did people calculate numerical values of transcendental and trigonometric functions?

You can often find discussions of this in old trigonometry texts. For example, see Chapter IX: Construction of Trigonometric Tables (pp. 82-93) in Isaac Todhunter’s Plane Trigonometry (1882).

As far as I am aware, approximating polynomials have usually been the most popular method since they have been understood for performing these calculations, both in computers and in the computation of tables. In ancient times, most trigonometric tables were constructed by physical measurement of carefully constructed triangles- there are stories of Greek mathematicians drawing strings across fields so that they could get more accurate measures. Some other methods, such as nomography and linear interpolation between known control points have been used with various degrees of success. Also, for computation of roots and logarithms, for a long time we were limited to the same simple brute force algorithms we teach children in grade school- it was the best option that was clear at the time.

It is also important to note that computation of mathematical tables typically employed many people working full working days, for many months.