How could you easily and accurately determine the glass
transition and melting temperature of a polymer?

You can use a thermal analysis called Differential Scanning Calorimetry (DSC)

Explanation:

The glass transition temperature (Tg) and melting temperature (Tm) are thermal properties of substances, such as polymers. You can determine them with a very low sample quantity by differential scanning calorimetry (DSC).

In DSC, there are two aluminium pans. In one pan, you put the sample, and the other pan is empty because is the reference pan. The pans are placed in a chamber of a DSC calorimeter and they are heated at a temperature range you select at a given speed (usually in ºC/min). The system register the difference of heat between the pans (sample pan and reference pan) and it shows you a plot of heat flow (in J/s) vs temperature.                                                                      

Tg is commonly detected as a step in the plot

Tm is usually an endothermic peak (upwards in the plot), because is a process in which heat is absorbed.