How to we work out the speed of an object at a certain point in its trajectory?

This depends on the path of the object and what information you have.
You could be looking at conservation of kinetic and gravitational potential energy. Total energy must be the same at all times. We can make a table to record the initial energies and then work them out at the required time. If kinetic energy and mass are known we can calculate velocity which is the speed in a certain direction: in this case the vertical direction as we have been looking at height in the gravitational potential energy.

Otherwise, if we have been given some angles, for example the direction a football is kicked we use the equations we have remembered to work this out more simply. Following a very simple and methodical approach and recording the things we know as the acronym SUVAT, this can become very simple.

The SUVAT equations are however just derived from conservation of energy with the added extra of resistance which comes into the equation as acceleration A. We could show this if you were interested.