It may feel like it sometimes, but the moon does not follow you individually! It is rotating round the earth, and completes one orbit every month. It will appear in different parts of the sky at different times in the month, and will also have a different appearance as it waxes and wanes depending on it’s angle to the sun. The light that comes from the moon is actually reflected sunlight. Around the new moon, it is sometimes possible to observe planetshine. This is when light reflecting from the earth reflects off the moon and makes it appear as though it is bathed in a soft light. It is really beautiful when you understand why it is happening!