Revealing mistake: The swords Jayden and Dekker use often bend when striking each other, revealing they are rubber blades and not metal.
Continuity mistake: All of the rangers were bestowed their powers when they were young children, including Emily's older sister. If the rangers were all selected when they were the same age, Emily should have been considerably younger than the rest of the team.
Suggested correction: I don't believe it was ever explained when the team was given their powers, so we can't know for sure that Emily would be the youngest. We learn during the season that the team was pre-selected and trained to become Rangers, but not when that took place.
Continuity mistake: Emily's eyes are a deep brown but in her flashbacks, they are blueish.
Plot hole: We learn the rangers were pre-selected to join the team and were in waiting until called to duty, but the yellow rangers sister just passed the legacy onto Emily, who was only a child.
Plot hole: This was the first episode of the series and we see that the rangers are unable to figure out how to properly operate their zords; but in Origins Part 1 (which takes place prior to this) they have no issues with the zords at all.
Continuity mistake: This is supposed to be the Rangers' first mission and we see Jayden use the Fire-Smasher mode of his spin sword. Jayden didn't learn how to use that feature until the third episode of the series, which takes place after this mission.