People can be surprised that in a course on pronunciation and speech training I rarely, if ever, refer to the phonetic alphabet. This is because I believe the phonetic alphabet has little relevance in improving pronunciation today.
The main reason for learning the phonetic alphabet is so that you can look up words in the dictionary and know their pronunciation by reading the phonetic script. But today, pronouncing dictionaries come with CDs with every pronunciation recorded and dictionaries online also have pronunciations you can listen to, usually with both American and British pronunciations.
If you are having trouble listening to the pronunciation of a word and are really concerned about it, you can always go back to the phonetic alphabet key at the beginning of the dictionary and figure it out. There's no reason to hog space in your head with the phonetic alphabet chart!
It is much more important (and much more fun) to learn pronunciation through fine tuning your listening and production skills. Although it isn't important for you to learn the phonetic alphabet, your trainer should know it like the back of his/her hand so she can make those sounds come to life and let you hear and understand the different nuances of sound. Make sure he/she has a background in either speech and language training or linguistics.
If you live in Singapore and are serious about changing your pronunciation and speaking more clearly, be sure to check out my newest program, the Speak Clearly 8-week Intensive. You can get more information and register here.