Jack's taxi is quite visibly battered with its side view mirrors broken off after he and the kids first elude the government agents. In the next (long view) shot, the mirrors are intact. A moment later (and in a close up shot), they are again broken.
Dr. Alex Friedman said to Jack Bruno that "closed-minded skeptics are always welcome". This was the wrong use of the word. Within the scientific circles from which Dr. Friedman comes, "skeptic" refers to somebody who rigorously uses the scientific method (the opposite would be "pseudoscientist"). Skeptics are, by definition, open-minded and Dr. Friedman herself is actually a skeptic, so that is most likely the word that she meant.
In the first chase scene, the taxi is boxed in on the left, right, and rear sides by black government vehicles. The rear vehicles distance changes from almost touching the bumper to a few hundred feet back and then back again.
After Seth stops the government vehicle the knocked out driver's sunglasses are hanging from his his forehead. While Jack and Burke exchange words, the driver is coming to and the glasses are still on. Right before Jack walks away, the driver's head is leaned back in the seat and the glasses are nowhere in sight.
When the cab is parked in the train tunnel, the blue light scans to within inches of the front bumper. The light fails to detect the car and in the next scene it is shown driving back through the tunnel in the other direction, without turning around.
When the RV is first seen outside of Las Vegas, the camera moves off the road and rotates to follow the RV. When it rotates far enough, the RV drives through dust where the camera was originally. The dust could only have been disturbed by a helicopter's rotor wash.
When "Jack Bruno"' is driving his taxi to the small country town, at least the two right side wheels are seriously out of alignment and both the front and rear bumpers are damaged. When they leave the town the Taxi wheel alinement's is perfect. Considering the garage mechanic only had one hour to do the repairs and it is very highly unlikely he would have color-coded bumpers for an out-of-town taxi, he would not have had enough time to complete the repairs as listed by Jack Bruno let alone the other repairs as seen as they leave town. However, there is nothing that suggests the mechanic didn't re-align the wheels first thing. It could have been that he started with the most obvious first and then moved to under the hood.
When talking about the Mustang on Jack's key chain which is hooked to the key in the ignition in the RV. The RV isn't Jack's, so it shouldn't even have been used at that time. However, given the sentimental meaning behind the key chain (as discussed in film) there's no reason he couldn't have removed it from his own key chain before handing the keys to the cab over.
When the kids are captured their street clothes are replaced with white scrubs and they are restrained on the operating tables. Later, when Bruno and Friedman rescues them the kids have their hand held alien tech with them, having them rather prominently on display in the lab, so they could have grabbed it on their way out.
When Jack and Dr. Alex gain control of the truck that is taking them away from Witch Mountain, the gear selector is clearly in the "Park" position even though they are "driving" (most obvious when Jack is in the driver's seat).
The goof item below may give away important plot points.
When the government team captures Sara and Seth and have them sedated, their clothing has been replaced with white clinical gowns. They are to be kept sedated and, as we learn, about to be dissected. So why would the government team put them in sneakers?