Continuity mistake: When Superman arrives at Lois' place she says "Don't fly away" and her hands are against her chest. A shot later her arms are extended.
Continuity mistake: When Johnathan Kent has a heart attack he passes by a harvest machine, walks several meters and collapses next to the house. In the close-up where he grabs his arm he's moved several meters behind and is next to the machine.
Continuity mistake: When Ms. Teschmacher saves Superman from drowning he is face up, but a shot later he is face down and swimming.
Continuity mistake: After Clark jumps over the railroad track there's a close-up of the train speeding by and it's obviously empty, no sign of the people that were previously seating inside.
Continuity mistake: In prison while Otis repeats what Luthor says he has a cop's hand on his shoulder or not, depending on the shot.
Continuity mistake: When Superman arrives at the broken railroad track the gap is huge, but when he places his body in between the tracks it is exactly Superman's height.
Visible crew/equipment: When the cocoon cradle crashes and the Kents pull-over, lights and bulky object (camera?) are reflected on the driver's door, perfectly visible when he opens the door. Seconds later, rigs and filming equipment are visible on the fender.
Continuity mistake: When baby Clark lifts up the truck, Martha's grip on his husband changes: In the close-up she grabs his chest, but in the wide angle she grabs his ribcage. This changes back and forth several times.
Other mistake: Clark does not have enough room beneath his clothes to have the suit, the cape and the boots.
Continuity mistake: After Superman gets soaked in Lex's pool he also gets his hair wet and messy, but when he flies up his hair is dry and combed and his clothes are dry.
Continuity mistake: While Lex and Otis are talking with the lieutenant and his crew at the bridge, the sun changes between shots.
Continuity mistake: In the extended version, where Martha is preparing breakfast, she places a box of Cheerios on the table. Then she looks out the window at Clark. The shot changes to an exterior shot of her looking out the window, but there is a pitcher of milk next to the box of Cheerios that was not there before.
Continuity mistake: While Lex is telling Superman his plans with the missiles at his home, he is touching his cane with both hands then only then both hands back again.
Deliberate mistake: At Lois' penthouse, before Superman says goodbye, the image is sharp and clear. Cuts to a close-up, and then check the next wide angle where he says goodnight and flies away: It's blurry and grainy because it's not live, but a screen with a previously shot scene to allow for the camera to turn left and, in a matter of seconds, no cuts at all, show Superman magically and quickly transformed into Clark Kent.
Revealing mistake: When the helicopter gets jammed on the white railing at the ceiling, Lois is replaced by an obvious dummy that dangles back and forth with no control.
Continuity mistake: While Lois and Superman are being mugged in the alley, Lois kicks the thief and protects herself leaning with her face against the wall and her arms widely spread. A frame later, when Clark grabs the bullet, her arms are close to her head.
Revealing mistake: As Air Force One enters the frame from the right, there appears to be a set of tracks faintly visible under it.
Revealing mistake: When the helicopter is hanging off the roof, a safety securing cable is attached to the fuselage to keep it in place.
Continuity mistake: When Superman lifts the plane during the storm, after the pilot asks what's going on, the copilot glances at the window grabbing its side. From the opposite angle he is away, repeating all movements.
Revealing mistake: When Superman and Lois begin their flight from her apartment, a pair of wires suspending each of them are visible, especially on the Blu-ray.
Answer: He wanted to see for himself if the stories were true. Some reporters tend to exaggerate the facts and if he had any other weaknesses. He couldn't be sure the kryptonite would work.
While I don't exactly disagree that Lex wanted to see for himself how invincible Superman is, I don't think that's the main reason why he did it. The bullets and fire were simply a charade to make it look like this was his security system. Keep in mind, he wanted Superman to enter his lair because the real trap was the Kryptonite that he had in the lead box.
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