1). To verify an identity means to prove that it is true. For example, an identity might be one equation or term that is equal to another and you have to prove that to be true using your knowledge of identities and your knowledge of simplification processes.
2). I have found that in verifying identities there are always the same strategies being repeated. These strategies include finding a GCF or a LCD, multiplying by the conjugate, separating fractions, converting into sine and cosine, using the zero product property, etc. Of these, I have found that converting into sine and cosine and separating fractions (only those with a monomial denominator) are two of the most recurrent and helpful strategies at approaching a problem. I also like to look at the answer when verifying to see if I can find any hints as to what to do to get there. Lastly, I also always look for things that might cancel each other out, since they can be very helpful.
3). I always ask myself when doing these problems "what can I do to change the problem in order to be able to use identities to get the answer?". Usually I look for anything that might hint toward a specific identity. For example, if I see cosine squared and sine squared I know I am going to be using a Pythagorean identity. Sometimes the problem is more complicated and you might have to alter it a bit before any identity jumps out in front of your face. A common example of this would be if you have to factor the problem first in order to get something that you can use.