Here are a few clues as to how to use those moon phases to your advantage:
New Moon (also known as "Dark of the Moon")
When the sky is void of a moon, it is like having a do-over. This is the perfect clean slate; spellcasters can use this phase to help remove unwanted elements in his or her life. It is perfect for cleansing and banishing negative energies. Remember, though, that when there's an empty space in your life, it must be filled. Try to do so with positive, healing energies and thoughts.
This is when the moon begins to grow. It's the perfect time to do spiritual work involving nurturing, growing, and creating. This phase is particularly useful for spells of personal power, protection, monetary abundance, and the improving of friendships and other relationships. Constructive and positive work will be enhanced during this time.
Between the new and full moon comes the First Quarter. This is the middle of the waxing phase, when the moon is "half cooked." As the moon continues to grow in size, you may find the best time for attracting and pulling positivity toward your life. Love spells and personal power spells flourish during this time.
The moon is getting bigger and approaching full-moon status. If you have any loose ends (such as unfinished spells or other spiritual work), this is the appropriate time to bring those to a natural conclusion. You may also wish to use this phase to plan ahead in what you'd like your future spellwork to include. Some also say that either Gibbous phase is an excellent time for divination and for doing protective spells or prayers.
A full moon is famous (or perhaps infamous?) for its mystical power. This is a perfect time for meditation and to gain strength. It's a wonderful time for practices that require the most powerful and potent magicks. If you're looking to use the power of the full moon in your spells, check the moon phase information for your area online and try not to wait until the full moon begins waning. (NOTE: Sometimes, a Blue Moon will occur; this is when a month has two full moons instead of just one. Use that extra full moon well. It can add a special power boost to your favorite full-moon spells!)
For those negative or annoying influences in your life, you will do best to banish them during the waning parts of the moon's phases. The waning gibbous, being the largest waning phase, can help with the really troublesome issues that you need to conquer. Just as with the waxing gibbous, it is always a good time for divination, as well.
As the moon begins to finish its cycle, you can use this time to seek balance. Focusing on eliminating negativity, while restoring positivity, and replacing the bad with the good, can be especially effective at this time of the month.
As you're waiting for the new moon to arrive, you can use this waning crescent for banishment of illness, depression, and to prevent harm against others. It is a cleansing and reflective time. You may also want to use this phase as inspiration to tie up loose ends and finish anything that you've been procrastinating.