Like virtually all Jedi, Mace deeply respected Yoda and was trained by him. He always obeyed the ancient Jedi Master, though he did occasionally disagree with him.
Mace had a healthy respect for Obi-Wan and Obi-Wan held Mace in high esteem for his mastery of Form VII. Despite their mutual respect for each other, however, Obi-Wan often disagreed with Mace's somewhat questionable decisions. This was because Obi-Wan was a moderate who followed the Jedi Code and Mace was an extremist who had a secret emotional attachment to the Republic.
Like most Jedi, Ahsoka held Mace in high esteem because of his mastery of Form VII, but he held her in low esteem because of her young age.
What bond they had between them was destroyed when Ahsoka was accused of treason; like most of the Jedi, Mace refused to believe in her innocence and coldly expelled her from the Jedi Order without a fair trial. After Ahsoka was proved to be innocent, Mace was unapologetic to her; instead of taking responsibility for his actions, he somewhat arrogantly claimed that her traumatic experience had been nothing more than a test given to her by the Force. Though he was visibly surprised when Ahsoka chose to leave the Jedi for good, Mace never showed any genuine shame, guilt or remorse over his unjust treatment towards her.
Mace first met Anakin when Qui-Gon Jinn brought him to the Jedi Temple. He was not very sympathetic with the nine year old, and like most members of the Jedi Council refused to allow Qui-Gon to train him due to Anakin's age and thus believed him too old to begin training as an apprentice.
Like Yoda, he was secretly afraid of Anakin, not because of what he was (that his connection to the Force was unusually powerful and strong), but because of what he could become in the future (an immensely powerful enemy of the Jedi Order and the Galactic Republic); by the time of the events of Attack of the Clones, Mace had outgrown his fear of Anakin.
After the invasion of Naboo was lifted, Mace reluctantly agreed to allow Anakin to be trained by Obi-Wan Kenobi, mainly because Obi-Wan had made it clear that he intended to train Anakin with or without the approval of the Jedi Council, as it was Qui-Gon's dying wish, but also because the Sith Order had come out of hiding and Anakin's potential made him someone who could one day defeat them.
Over the years, Mace and Anakin came to respect each other fairly well due to their many similarities; both were born in the Outer Rim, both had secret emotional attachments, and both were hardliners who mastered particularly aggressive lightsaber styles (Anakin mastered both variants of Form V: Shein and Djem So, while Mace mastered both variants of Form VII: Juyo and Vaapad). However, they didn't always agree with each other and were not necessarily friends.
The pair's respect for each other was severely shaken when Ahsoka was accused of treason; Anakin saw right away that Ahsoka had been framed, but Mace refused to even see this and urged that Anakin be left out of the matter because of the emotional bond between him and Ahsoka, but was undermined by Yoda and Obi-Wan. When Tarkin demanded that Ahsoka be expelled from the Jedi Order and turned over to the Republic military for her so-called treason, Mace urged them to give in. After a short Council meeting, Mace coldly had Ahsoka expelled from the Jedi Order. Anakin saw right away the meeting was just a formality to disguise how the Council was casting Ahsoka out for the sake of self-interest and was disgusted by Mace's shamelessness. After Ahsoka left for good, Anakin secretly developed a grudge against Mace.
As the Clone War dragged on, Mace became increasingly mistrustful of Anakin, believing he was unstable, unpredictable and unreliable. Anakin's dislike for Mace was intensified when the latter rather coldly denied him the rank of Jedi Master when Anakin was placed on the Jedi Council, something Anakin perceived as an insult. Mace also ordered him, via Obi-Wan, to spy on Palpatine, whom Anakin considered a friend and mentor, further fueling Anakin's anger. When Anakin informed him of Palpatine's true Sith identity, Mace quickly accepted the truth, but refused to allow him to accompany him in confronting Palpatine, believing he was too much of a risk.
After Mace subdued Palpatine, he resolved to execute the Dark Lord of the Sith, believing he was too dangerous to be left alive due to his near-dictatorial power in the Senate. Anakin pleaded with Mace to spare Palpatine, pointing out that killing an unarmed opponent was not the Jedi way, but Mace, too focused on the Dark Lord of the Sith and blinded by his emotional attachment to the Republic, ignored him and prepared to strike. However, Anakin stopped Mace by cutting off the hand wielding his lightsaber, allowing Palpatine to kill him. Despite his dislike for Mace, Anakin was horrified by what he had done and showed at least some remorse over his part in Mace's demise, but knew there was no turning back and reluctantly pledged himself to the Dark side.
At first, Mace respected and admired Palpatine for his clean slate. When Dooku revealed the existence of Darth Sidious and that the Dark Lord of the Sith had hundreds of Senators under his control, Mace, unlike Yoda, took it seriously and vowed to keep a closer eye on the Galactic Senate. As Palpatine amassed near-dictatorial powers, Mace began to realize he was not what he seemed. When the CIS invaded Coruscant, Mace realized that it was an inside job and that Dooku was telling the truth.
Mace later had Anakin spy on Palpatine, and Anakin later revealed that Palpatine and Darth Sidious were in fact the same person. Shocked and startled by this terrible revelation, Mace immediately confronted the Dark Lord and just apparently barely managed to defeat him with a combination of his mastery of Vaapad and his shatter-point ability. Deciding that he was too dangerous to be kept alive, Mace prepared to execute the Dark Lord of the Sith, only to have his sword arm severed by an emotionally insecure and desperate Anakin, allowing Palpatine to kill him.