I have M(T) and M(H) data of magnetic samples and I would like to calculate the thermal effects like specific heat or heat capacity and entropy change in an easy and direct way.
The name of the game is Maxwell relations. With M(T) you have the the temperature dependence of the derivative of free energy versus magnetic field. Supposing no intermediate phase transitions, the free energy is analytical over that temperature range so you also have the magnetic field dependence of the entropy (derivative of free energy versus temperature): d2 F/(dT dM) = d2 F / (dM dT). With that information you can deduce the entropy change with magnetic field.
Note that one only obtains changes, no absolute values.
Entropy: A concept that is not a physical quantity
https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity