Suzaku was always Lelouch's enemy; they had the same goal, but different ways.
But when they met and killed Lelouch's father, he changed sides and became Lelouch's Knight. Why did this happen? At that point, Lelouch had the Geass in both eyes. Could he now manipulate people more often? Or did he just agree with Lelouch's plan to cheat the whole world?
TL;DR: He never changed sides.
Suzaku's allegiance remained with the Britannia Empire the whole series. After the events in C world that culminated with the Emperor's death, Lelouch became the new Emperor, and it was only logical that Suzaku would serve Britannia's Emperor, Lelouch.
Both Lelouch and Suzaku had the same goal: to restore Britannia and end the corruption. Suzaku wanted to do it by working WITHIN the system, Lelouch would do it by the fastest means available. That is why they were at odds before the Emperor's death. Once Lelouch became part of the system, that rivalry ended.
Also, Lelouch had already used Geass on Suzaku in the first season, by ordering him
To Live!
that was triggered anytime Suzaku was at a risk of death, most of the time improving his combat edge.
Lelouch could use Geass more than once on an individual, only if Orange-kun used his Geass Canceler between uses. Happened only once in the whole series, with Shirley: he used it once to eraser her memory, then Orange-kun wiped that effect, and later while she was dying on Lelouch hands, he commanded her to live, but she was unable to carry that and died anyway.
Agreeing to the Zero Requiem plan suited Suzaku on several levels. He would
Near the end of the series it is explained how Lelouch came up with the Zero Requiem; that being, the idea of creating a focal point of the entire world's hatred and then having that point be destroyed thus "Breaking the chains of hatred" furthermore he would make this focal point himself. Thus, by giving the entire world a single target, they would be united in their fight against the tyrannical leader he became.
Through his death, he could facilitate a peace throughout the world. It is this idea that led Suzaku to join Lelouch, this also coincides with the fact that Lelouch would also be atoning for his sins through his death, one sin in particular being the death of Euphemia li Britannia. This fact also contributes to Suzaku joining Lelouch as he would gain both justice for Euphemia as well as being able to bring about lasting peace.
I think you guys are missing the point. By the end of episode 21, I believe that Suzaku had forgiven Lelouch for what he had done. There are many reasons why he did this.
Firstly, in episode 'Assassin from the Past', Shirley reveals that she has forgiven Lelouch for something that he had done to her an urges Suzaku to do the same. Also, by the end of the episode, Suzaku realises that both Euphemia and Shirley knew who Zero really was, and yet they both kept his secret to the very end, even though revealing his identity would have prevented both their deaths. This leads Suzaku to realise that Zero may not be as bad as he thought.
Furthermore, in 'One Million Miracles' while contemplating what to do, Suzaku comments that both Euphemia and Nunnally wanted to forgive Zero for his sins. Although it is not enough at the time, it does create doubts within Suzaku which help lead to his changing sides.
Moreover, in 'A Taste of Humiliation', Suzaku confronts Lelouch about his past sins. Lelouch, not caring about himself and wanting punishment to atone for his sins, lies about everything he's asked about. Suzaku, having experienced this already, realises that he's lying, and perhaps finally understands that Lelouch didn't want to kill Euphemia or anyone else. This allows him to open up to the prospect of forgiving Lelouch, though he doesn't at that particular moment. This moment, in particular, was paramount in Suzaku's joining sides with Lelouch.
Finally, after the main events of the 'The Ragnarök Connection', Suzaku, once again, confronts Lelouch over Euphemia's death, but Lelouch challenges him and says that nothing is unforgivable. (Since Lelouch never blamed Suzaku for killing his father and that he was forgiven by those closest to him, Suzaku found it in his heart to forgive Lelouch, too.) This, coupled with that fact that Suzaku had realised that Lelouch was now fighting for all humanity, is the final turning point in the friendship/enmity between Lelouch and Suzaku.
PS. For those with any doubts that Suzaku had forgiven Lelouch by the end of the series need only be reminded that Suzaku was weeping when he killed Lelouch.
As far as we know, it's true that their ideals were fundamentally the same:
To change the corruption of the Britannian Empire, each had his own way to do it.
As for Suzaku, he didn't agree with Lelouch's way because he wants to change and improve the Empire from within, to show that his father's death was not in vain.
This is because, as Suzaku claims, achieving results by wrong or illegal methods is meaningless.
The reason Suzaku chose to join Lelouch is because after the incident with the death of Euphemia, Suzaku's only motive to keep going basically was to protect Nunnally as long as Lelouch will pay for what he did to him.
