When receiving a new transformer , what are the tests that will be applied on it ?

Install and check operation of equipment, measure the voltage, low could be excessive load(another problem).
You could also check current to compare against known, or a good circuit current.
Edit, I just confirmed this is for a test, Answer may be different than real life.
An example is a transformer for an Air Handler or Alarm panel. If it is bad, you replace, if the equipment works, trsndformer test good?

Beyond checking for visible damage, and insuring all packing and shipping materials are removed, before energizing any transformer, Doble and TTR tests are performed:

Doble Engineering

Megger

Dear Jaguar,

Upon receipt of new transformer, very first test is to be carried out is magnetic balance test. This is to ensure that there no core / winding damages during transportation of the equipment. Then, ratio test to ensure that there is no dry soldering if any inside of the tank. Finally, IR check with appropriate instrument.

Vector group and other tests are not mendatory upon receipt of the machine if these tests are carried out at manufacturers works. You may contact me on my email id sathe1963 gmail.com