Short answer
A new Moon (Amavasya) opens the lunar month and suits quietly setting an intention; a full Moon (Purnima, or Pournami in Tamil) is the peak and suits finishing, celebrating and letting go. What it means for you depends on which sign the Moon is in, counted from your own Moon sign. The Panchangam shows this week's dates for your city.
How ChiChakra reads this
ChiChakra counts the sign of each new or full Moon from your Moon sign (Gochara). A full Moon in your own sign tends to feel emotionally loud; one in the 8th sign from yours falls on your Chandrashtama days, a time for lighter plans. The Moon spends only about 2.25 days in each sign, so these effects are brief.
A simple practice
On the new Moon, write down one thing you want to begin this lunar month, small and specific. Amavasya is also a traditional day for remembering ancestors, and many families avoid big launches on it.
On the full Moon, look back at that note. What moved, and what can you drop? Purnima is a traditional day for prayer, fasting or a temple visit.
Repeat this for three months and your own pattern will show more clearly than any forecast can.
Good to know
Western sites give the same phase times, but their sign names usually run one sign ahead of the Vedic ones.