Between TPM and AOTC, Palpatine was able to persuade Count Dooku to join his fight for a "more peaceful and stronger Republic". Dooku and his friend Master Sifo Dyas was tired of the weakened Republic.
In the mean time, Sifo Dyas ordered a batch of Clone Armies from Kamino to help the Republic as he as forseen that the Sith were gaining powers.
Palpatine instructed Count Dooku to erase Kamino from the Jedi Archives and also to murder Sifo Dyas before joining him for more Sith Trainings. The murder of Sifo Dyas was essential as Palpatine wanted to manipulate with the Clones by programming them to absorb
Spoiler:
Order 66 which was to murder all the Jedis which led them in the Clone Wars in AOTC and ROTS
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In AOTC, Count Dooku created the Separatist Movement and split the Republic into 2 fractions. Palpatine was determined to being an end to this Separatist movement. Concurrently, Palpatine made Anakin Skywalker aware of his extraordinary Jedi skills, and became Anakin's closest friend on Coruscant after Obi-Wan.
Meanwhile, Palpatine officially "learnt about" the Clone Armies being secretly made at Kamino through Obi-Wan's report from Geonosis. Taking advantage of the absense of Senator Padme Amidala, Mas Amedda suggested emergency powers to be given to Palpatine to quickly commission the Clone Armies for the use of the Republic. He directed the thought that "Senator Amidala was the brave Senator who would have suggest this Emergency Power" to Jar Jar Binks, who was replacing Padme as Senator for the time being. Jar Jar Binks, not knowing the full impact of this emergency powers, requested for full power to the Chancellor during the midst of an Emergency Senate Session. The Senate approved it immediately. And the Clone Armies were dispatched to Geonosis and subsequently to the rest of the Galaxy to stop the Separatist Movements. And this war was known as the Clone Wars.