I would have taken out Qui Gon's role all together. Have him as an extra, maybe, or a small supporting character, but the film from the get-go ought to be focussed on the relationship between Obi-Wan and Anakin in order to make Anakin's ultimate betrayal that much more tragic.
We don't need Anakin's whole life-story, and if we do, include it as a series of memories/flashbacks rather than a whole film in and of itself. Start the prequels in the midst of the Clone Wars. The Jedi are already a dying order, with none of the pomp and ceremony they are shown to have. They are a bit like the Knights Hospitaller, relatively isolated but a branch of the military with some degree of independence from the Republic. Have it be known that their downfall was a direct result of becoming too involved in the Republic's political machinations. With the Clone Wars going on, they have been forced to take a more active role.
In Episode I, Anakin is already a Jedi under the tutelage of Obi-Wan Kenobi. He is young, brash, handsome, and optimistic. Obi-Wan is of a sterner sort, doing things more "by the book" and adhering more closely to older, more restrictive doctrines. He disapproves of Anakin's demeanour. Anakin relishes in the heroism, and the love they receive from the people they help save. In Episode I, Anakin and Obi-Wan are involved in a mission to Alderaan, where they need to rescue King Organa and his family, which includes some of their staff, such as the young, beautiful, alluring Padme Amidala. While fleeing the Clone Armies, Anakin and Padme grow closer and closer, until Anakin does something wholly questionable near the end of the movie in his defence of her. Of course, this only comes as a partial surprise. While fleeing from the clones, we see snippets of the horrors of war, horrors the Jedi weren't directly involved in due to their stance as an independent humanitarian force.
So Episode I ends with a darker twist on Anakin's otherwise likeable personality, done in the name of the woman he has grown to love. There are no Sith in this film, not directly. They are alluded to near constantly with small scenes told from Palpatine's perspective. The Sith are presented as a cult, almost Satanic in nature, the antithesis to the Jedi, born almost completely out of the Jedi Order itself. We see them as we see the Templars in Assassin's Creed; they aren't obviously evil, but we know they wish to be the guiding hand in the galaxy.
Episode II reveals more about this Sith threat and introduces Darth Maul. We get to see the Sith as liberators rather than peace-keepers. They sweep in and restore order to an otherwise chaotic galaxy, which Anakin takes not of. In this film, while he pursues a secretive relationship with Padme, we get to see two different viewpoints that both influence Anakin; that of his lover, and that of his mentor. Padme openly wonders if the Jedi are still relevant, or if they should hand the reigns over to the Sith. In this version she isn't an idealistic politican after all, but just a household staff member to the Alderaan royal family. Obi-Wan, conversely, sees the Sith as an ultimate evil, and recalls the times of the past where the Sith attempted to seize power.
Anakin is conflicted. On the one hand, he has been raised as a Jedi and he still has some hope that the Jedi Order is on the right path, but on the other he sees the cold efficiency of the Sith to be a more straightforward method of ensuring peace. This film ends with a brutal battle between Anakin, Obi-Wan, and Darth Maul, kind of like the battle at the beginning of OTL Episode III where Anakin and Obi-Wan fight Count Dooku. It goes much the same way. Maul taunts Anakin by drawing out his anger and frustration, and mocks the Jedi. He disables Obi-Wan and Anakin faces him alone. This is still all done before the eyes of Chancellor Palpatine. As with Episode III, Anakin gets Maul on his knees, and Palpatine urges Anakin to kill Maul, so Anakin does.
Episode III: A few years down the line, the war hasn't gotten any better. Padme is pregnant and Anakin is more desperate than ever to provide a safe and stable world for his family to live in. He's seen children die. He's seen families torn apart. He wants to avoid that for his own, and he doubts his power to protect them. The Jedi hear of Sith cultists popping up in different places and now it is their primary goal to destroy them for the sake of balance in the Force. Anakin disagrees with this; let the Sith do what they're doing, and the Jedi should stay true to their efforts to save the people who suffer the most in the war. He follows, orders though. The difficulty comes when Anakin and Obi-Wan join forces with Mace Windu to combat a large group of Sith, but when they arrive they learn that one of Anakin's old friends from the Jedi Academy days has joined the Sith. Anakin can't bring himself to kill his friend, considering the friend is still the same. Same personality, same drive, same easy-going smile. Windu kills him when diplomacy fails.
Anakin is summoned by Palpatine, who engineered the whole thing. Anakin freely expresses his frustrations to Palpatine, who throughout the series has been something of a grandfatherly figure to Anakin. Palpatine somehow knows about Padme's pregnancy and uses this to convince Anakin to steel himself and "do what must be done". Anakin deduces that Palps is a Sith and Palpatine freely admits it. He begs Anakin to join him. Anakin refuses, struggling to do the right thing, and he goes to Windu. Windu assembles a force of Jedi and they attack Palpatine. All except Windu are slaughtered, just as before. Palpatine is on the ropes now, just as in OTL. He begs Anakin to see the truth of the Jedi and sense the corruption in their Order and in the Republic. Anakin makes a snap decision, and kills Mace Windu, just as before. Only this time, it is less directly to do with Padme, and therefore less cheesy.
The rest goes more or less as OTL, except with one big difference. Instead of killing the Clone Army leaders on Mustafar, Anakin is sent as an emissary of "Lord Sidious" to a banquet being held by the leaders and their families (including women and children). Anakin slaughters all of them. Order 66 is executed as OTL and the Jedi are systematically hunted down and destroyed. Obi-Wan meets with Yoda (who would have appeared intermittently throughout the series) and they try to figure out what to do. Obi-Wan meets with Padme and tries to convince her to see that Anakin has changed. Padme refuses, however. As before, she leads him to Anakin by accident, and as before Anakin hurts her (although it would be far more convincing). Cue the epic final battle between Anakin and Obi-Wan.
I think the over-all plot of Episode III at least was fine, as was the idea of a stretched-thin Jedi Order caught up in a war. It's the smaller details that need changing.