It would be a waste to do this experiment before you have done it in class. You may be able to pick it up from the textbook, but give your teacher a chance to cover it in class.
A short circuit test is for finding the copper losses in a transformer. A very small voltage is applied to the primary of the transformer and the voltage is increased until the current in the short circuit is equal to the maximum load current of the transformer.
Then measure the power in. This is mostly copper loss because very there is very little magnetic core flux flowing to get this output.
The open circuit test tests for core losses or iron losses. Because there is little current flowing in the wires of the transformer, most of the power used is iron losses like eddy current loss.
Testing a known type of transformer for faults will be mostly a matter of comparing the results of the above tests against those of a known "good" transformer.
Testing an unknown transformer is more difficult and may be a matter of judgement.
Obviously, if it is known to run very hot or to blow fuses, then it is probably faulty.
If it runs hot without any load, then it is possibly faulty, or it may be a poor design.
If it draws a large primary current without any load (say, maybe more than 10 % of the rated full load primary current) then it may be a poor design or it may be faulty.
You might like to read the Wikipedia article about this:
http://en.wikipedia.org/wiki/Short_circuit_test
Here is a bit of it:
The purpose of short circuit test is to determine the series branch parameters of the equivalent circuit. as the name suggests, in this test primary applied voltage, the current and power input are measured keeping the secondary terminals short circuited. Let these values be Vsc, Isc and Wsc respectively.
The supply voltage required to circulate rated current through the transformer is usually very small and is of the order of a few percent of the nominal voltage. The excitation current which is only 1% or less even at rated voltage becomes negligibly small during this test and hence is neglected. The shunt branch is thus assumed to be absent. Wsc is the sum of the copper losses in primary and secondary put together. The reactive power consumed is that absorbed by the leakage reactance of the two windings.
