Friends and Allies
Qui-Gon was Dooku's apprentice. Dooku loved Qui-Gon like a son due to their many similarities; both were individualists, idealists, and both had knowledge of the Force that was unique. When Qui-Gon passed the Jedi Trials and became a Jedi Knight, Dooku told him not to trust others because his need to reach out to others left him vulnerable to betrayal. When Qui-Gon was killed by Darth Maul, Dooku was so saddened that Qui-Gon's death lead to Dooku's transformation into the Sith Lord Darth Tyranus. He would later provoke the memory against Obi-Wan Kenobi, Jinn's apprentice.
Like Asajj and Savage, Grevious was trained personally by Dooku. Dooku taught Grievous the art of lightsaber combat, making him formidable as an opponent to the Jedi. After Asajj's plot against him was revealed, Dooku dispatched Grevious to Dathomir to kill the Nightsisters, where the Kaleesh warlord performed magnificently. After Dooku was killed, Grevious assumed leadership of the CIS.
Yoda trained Dooku as both a young boy and as his apprentice. The two became enemies after Dooku turned to the dark side.
Obi-Wan was Dooku's polar opposite. Dooku hoped to have Obi-Wan as his Sith apprentice and went as far as to reveal Sidious's existence to convince him to join him. Obi-Wan didn't believe him and refused, resulting in the two being enemies ever since. Despite this, however, Dooku had a healthy respect for Obi-Wan, considering him a worthy adversary.
When Dooku first encountered Asajj, he was impressed by her skills such as her mastery of Jar'Kai, but was not impressed by her declaration to be a Sith. He told her that she lacked the vital qualities that make one a Sith and proved his point by demonstrating his mastery of the Sith Arts. Afterwards, he made her his apprentice and trained her in skills he himself excelled in such as Makashi.
However, Palpatine became concerned about her growing power and coldly ordered Dooku to kill her. Out of self-interest, Dooku betrayed her and left her for dead. However, Asajj survived and the two have been mortal enemies ever since. Dooku eventually killed Ventress with Force lightning when she threw herself between him and her love interest, the Jedi Quinlan Vos.
After he abandoned Asajj, Dooku took Savage Opress, the younger brother of Darth Maul, as his new apprentice. He was impressed by the Nightbrother's power and taught him how to refine it. Unknown to Dooku, Savage was really a sleeper agent who was placed by Dooku's side to avenge the betrayals of both Asajj and Maul. This was revealed when Asajj prematurely revealed herself. Fortunately for Dooku, he was able to hold the pair off by using Savage's lack of skill against them.
Even though they were both trained by Palpatine, Dooku and his predecessor, Maul have been enemies every since his mind was restored by Mother Talzin. This is because the two hated each other. Dooku despises Maul for being a threat to him and Maul hated Dooku for being his replacement. They also didn't regard each other as a true Sith; Dooku saw Maul as nothing more than a trained animal and Maul saw Dooku as a Sith pretender.
Dooku had very little respect for the young Jedi Knight, despite his bravery and unbelievably strong connection to the Force. Meanwhile, Anakin developed a grudge against the ex-Jedi Master/Sith Lord for cutting off his right arm.
During their final duel, Dooku learned why Palpatine was so interested in Anakin; Anakin was half-Sith already and he didn't even know it. When Palpatine ordered Anakin to kill him, Dooku finally realized he was never a true Sith as far as Palpatine was concerned, but merely an instrument to help him turn Anakin to the dark side. Due to their long history as enemies, Dooku could be considered the archenemy of Anakin, similar to Obi-Wan and Darth Maul.
Moments before he was killed by Anakin; Dooku finally come to respect him, and (silently) acknowledged Anakin as the finest lightsaber duelist that he had ever seen.