Things You Can't Outrun - S1-E3
Other mistake: When Cisco and Cailtin call Barry to tell him about the man with the gun, Cailtin says he has a gun. Cut to the cops pursuing him, the perp shoots at the cops, and one of them seems surprised that he has a gun. There's no way Caitlin could have known about the gun without the cop knowing (which would presumably have been relayed over police communications).
Other mistake: In this episode, Hera's ears are shown for the first time thanks to her civilian disguise. As for all Twi'lek women, they're shaped like cones. However, they are too large to fit under the ear coverings of the pilot's cap she usually wears, when they've been shown on the few occasions she hasn't been wearing goggles or a pilot's visor.