The AlphabetFarsi uses the Farsi script, which is similar to Arabic script but includes a few additional letters. Even if you only want to learn a few phrases to be polite, I recommend learning the script so that you can decode signs and use a dictionary. You may find my general tips on learning another alphabet useful. I also recommend the following 3 websites aimed at teaching the alphabet to children, teens, and adults - all of which are excellent introductions.
Simple Words and PhrasesTo move beyond recognizing the letters to whole words, I highly recommend the excellent book Your First 100 Words in Persian, which includes puzzles, games and flash cards to help you learn some basic Persian. You can start with my collection of useful words and phrases in Persian. You may also be interested in our Middle Eastern word of the week, which is available in our newsletter. If your primary interest is learning a few phrases to be friendly or find your way around when traveling, you probably want a phrase book. I particularly like Travel Talk Farsi which includes your choice of a CD or cassette with about 300 key words, a pocket-size cheat sheet of the audio, and a copy of Lonely Planet's Farsi (Persian) Phrasebook. The later is also available separately and includes a dictionary, well-written phrasebook and grammar notes. I particularly like the way Lonely Planet phrase books are organized to make them convenient to use for pass and point communications in various standard situations (restaurants, shopping, etc). I would also recommend Language/30: Persian. This is also a combination audio and text product. It includes CD's or cassettes and a small booklet with the words and phrases covered on the audio. The Language/30 product includes more phrases on the audio, but fewer total phrases, so the best product for you probably depends on how you learn and how motivated you are.
Beyond PhrasesIf you want to go beyond greetings and phrases toward true language competency, I really haven't seen anything better than Hassan H's website Easy Persian. This site was mentioned earlier as a source for learning the alphabet. This site includes audio, text, exercises, quizzes and even forums and chat rooms to learn Persian. It is through and well-written and all around one of the best language learning resources I've seen for any language.
DictionariesA good dictionary provides enough context that you can avoid making mistakes like saying "rotate correct" when you mean "turn right." I haven't found a good paper dictionary yet, but this level of detail is exactly why I like the ECTACO online and electronic dictionaries.
Enjoy learning Persian and remember that any time spent studying a language improves your understanding of that language and the people who speak it.