Look for the moon each night and keep track of how it changes. It may not be possible to see it every night due to clouds or forgetting, but try to do it as often as possible. You can just draw a series of circles in your nature journal, or here is a printable page to help you keep track: Moon Journal. Note the date and time for each moon picture. Perhaps you may want to draw one big moon picture on the date of the full moon and note the craters and shapes you can see.
You can find out when the moon is, and when the moon will rise and set in your part of the world here: Moonrise and Moonset.