Shemini Atzeret and Simchat Torah actually have no inherent mitzvot. They are observed by beginning the recitation of the prayer for rain (tefilat geshem), singing and dancing with the Torah, completing the yearly cycle of Torah readings and beginning the next cycle and refraining from work. 

The prayer for rain (tefilat geshem) is recited beginning on Shemini Atzeret since the day marks the start of the rainy season in Israel. 

Until Passover, the phrase “masheev ha’ruach u’moreed hagashem,” “Who causes the wind to blow and the rain to fall” is recited in the Amidah prayer.

On the first day of Passover, we begin to pray for dew (tefilat tal).

Both prayers coincide with the changing of the agricultural seasons in ancient Israel. We pray for rain at the beginning of rainy season in Israel and for dew before summertime, which is traditionally long, dry and hot. 

In times when agricultural success was determined by weather alone, a tiny bit of dew falling in the morning could mark the difference between a successful harvest season and a poor one.

Both the prayer for rain and the corresponding prayer for dew are based on the idea that everything in nature is a blessing from God, especially rain and dew. 

There are six parts of the prayer for rain, each of which refers to events involving water in the lives of Abraham, Isaac, Jacob, Moses, Aaron, and the Twelve Tribes.

The full Hebrew prayer for rain can be found here.

The english, which has been translated by My Jewish Learning, is as follows:

Our God and God of our ancestors:

Remember Abraham who flowed to You like water.

You blessed him like a tree planted by streams of water.

You rescued him from fire and water.

He passed Your test by planting good deeds by every source of water.

For Abraham’s sake, do not keep back water.

Remember Isaac, whose birth was foretold when Abraham offered the angels a little water.

You asked his father to spill his blood like water.

In the desert Isaac dug and found wells of water.

For Isaac’s sake, do not keep back water.

Remember Jacob, who crossed the Jordan’s water.

He bravely rolled the stone off the mouth of the well of water.

He wrestled with an angel made of fire and water,

And therefore You promised to be with him through fire and water.

For Jacob’s sake do not keep back water.

Remember Moses, who was drawn in a reed basket out of the Nile’s water.

Who helped Jethro’s daughters: He drew water and gave the sheep water.

He struck the rock and out came water.

For Moses’ sake do not hold back water!

Remember Aaron, the High Priest, who, on Yom Kippur, washed himself five times with water,

He prayed and was sprinkled with purifying water,

He kept apart from a people who were as unstable as water.

For Aaron’s sake do not hold back water.

Remember the Twelve Tribes whom

You brought through the divided waters;

For whom You sweetened bitter water;

Their descendants’ blood was spilled like water.

Turn to us, God, who are surrounded by troubles like water.

For the Jewish people’s sake, do not hold back water.

You are Adonai, our God

Who causes the wind to blow and the rain to fall.

For blessing and not for curse. Amen.

For life and not for death. Amen.

For plenty and not for lack. Amen.