I would have Dr. Connors be the first villain. Like someone suggested, in the beginning Peter could already be Spider-Man facing off against some bank-robbers and I would have it where they try to escape via helicopter like we saw in 2001 trailer for the first movie, but they get caught in his webs like a fish in a net.
From there we get introduced to Pete at school. He is a loner--- his only friends are Gwen and Harry. Harry sticks up for him against Flash Thompson and the rest of the bullies, but feels that he's missing out on the popularity he should and could be having because he has Pete to look after/tag along. Gwen secretly loves Peter/Harry secretly loves Gwen.
Pete, being the head of his class, is asked to work after school for Dr. Connors. A friendship is developed between the two men and Connors tells Peter about the formula he's been creating that can regrow a human limb. Peter thinks it sounds dangerous and warns Curt of the variables and possibilities that occur if he does indeed test it. Curt doesn't listen, he tests it, and becomes The Lizard.
Meanwhile, Tombstone or Kingpin could be a secondary villain like Falcone in "BB." I would have Norman Osborne being a shadowy character. We meet him through Harry but we don't quite know what he's doing, who he's involved with, etc. because he doesn't want anyone in on it and it's all hushed-hush and behind close doors. At the end, he would reveal himself as the man controlling Tombstone and supplying the mob with weaponry. Perhaps, the Enforcers could be thrown in, but just in name only.
In movie 2, I'd have where we learn more about what Norman is doing. Vulture could be the villain this time. He gets revenge on Osborne, Osborne gets found out that he was helping Tombstone, and in his last case scenario he tests the Goblin serum, and flees Oscorp. Harry defends his dad saying it's all rumors and that there is no proof. Pete and Gwen's love blossoms. Spider-Man defeats Vulture but GG shows up, kills Toomes, and takes Gwen hostage. Spider-Man chases GG to the bridge, GG drops her, she dies, and Spidey beats GG to a bloody pulp, but soon sees this isn't how he should be or react. He lets GG go but GG with a last ounce of strength tries to kill him but of course... we know what happens.
Movie 3, I'd bring in some new villains like Kraven, Electro, Mysterio, or Chameleon.
Movie 4- Doc Ock.