Is it possible to find out the distance between the moon and earth using only basic geometry and trigonometry?

No. You will need a base to your triangle which involves moving a known distance from which you would have to accurately measure the angles of the other two sides of your triangle. Whilst you are moving the Moon to would be moving. You really need to rely on established facts about the Earth's diameter, the Moon's orbit etc, in order to create your triangle and use trigonometry to calculate the Moon's distance which is a variable anyway.