When Billy calls to his friends from the water, Trini (Becky G) is seen laughing and turning around. The angle changes and she's seen standing facing forward with a flat face, while the next angle change has her laughing and turned around again.
In the scene when we first see the morphed rangers from the back walking towards the cliff edge to see the gold monster marching towards the city, their helmets from the back can be seen having two pieces, but when the camera shows them from the side with their faces visible, the helmets are one piece.
When Rita goes to the jewellery store she tells the assistant to hand over her necklace. The assistant starts to take it off but in the next shot it is no longer round her neck & Rita is already holding it.
When Trini cliff jumps as the rangers are discovering their powers, Kim can be seen and heard beginning to say "What is wrong..." as the other rangers are shown running up to join her. The camera then focuses on Kim and she says the full phrase "What is wrong with you!" over again and in a different tone of voice.
In the scene right after sam's brother flips the truck over, the camera shoots from red ranger's view, we can see the steveston whale watch shuttle bus facing south, but in the next shot we see the bus flipped facing north, clearly indicating it was shot on two different days.
At the beginning of the film, the caption reads 'PLANET EARTH: CENOZOIC ERA', and what appear to be pterosaurs can be seen flying in the background. The Cenozoic is the current geological era, and pterosaurs went extinct in the Mesozoic.
When the teens are swimming through the subterranean lake, while they are nominally swimming down/sideways towards the cave with the ship, the bubbles they're exhaling move "forward" from their face, revealing that they're really facing up, and the effect is being accomplished by altering the camera orientation.
After finding the power coins the rangers have super strength without morphing. However when Rita drops Billy into the water at the docks the rangers have to work together to pull him out of the water and carry his body back to the ship.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
When Billy drowns the team never attempted to perform CPR. If CPR did work on Billy then Zordon may have been free. Billy's "death" was caused by more than drowning, including Rita's magic (you can clearly see a deadly magic effect on his body) and the abrupt, forced plunge into the water. Furthermore, his neck could have been broken by the chain. While the drowning was certainly part what happened, it was not the only thing that caused him to die. CPR could have been a solution for the drowning, but certainly not the other factors that resulted in his death. Furthermore, Zordon being free would not have mattered in the long run; he even says that "there can only be one Red Ranger."