During the first scene on the bridge, when Troi senses the Q-being, Picard refers to her as 'Commander'. But later on the planet's surface, Riker tells her, "You have your orders, Lieutenant." Actually, her rank through most of the series is that of a lieutenant commander, and although the usual form of address would be 'Commander', it appears that either appellation can be used for that rank.
Deanna Troi tells Gropler Zorn "I am only half Betazoid...my father was a *Starfleet officer*," as if that alone rules out the possibility of him being Betazoid. Which turns out not to be the case, as we later meet plenty of full blooded Betazoids who are also Starfleet officers, throughout this and other Trek series.
Riker tells Data that he checked Data's records. Apparently he didn't read them very well, because he soon is surprised to learn that Data's rank is real, not honorary, and that he graduated from Starfleet Academy.
Wesley is dripping wet when talking to Captain Picard outside the holodeck. As holographic matter cannot exist outside of the holodeck, the water should have disappeared once he stepped through the door.
The computer, via the information panels along the corridors, is directing Riker to the holodeck to find Data. Riker is walking down a corridor when the computer announces, "The next hatchway on your right." The information display panel is on the bulkhead to Riker's right. Riker stops and pivots left (not right) to the holodeck door.
Even though he is a commander in Starfleet, Riker doesn't seem to know that he can use his communicator badge to speak to anyone within communicator range who is wearing his or her badge. So instead of asking a random crew member where Data is, Riker should simply touch his communicator and ask Data where he is.
When Q appears on the bridge as a 20th-century US military officer, he is wearing the rank insignia of a Captain, probably in an attempt to portray Picard's rank. However, in the branch it is supposed to depict, Captain is equivalent to a full Lieutenant, since Starfleet uses the Navy's rank system, not the Marines' or Army's.
As Riker leaves Farpoint with Data to investigate the explosions which are coming from the Bandi village, Troi yells out, 'wait! If you should be hurt', Riker responds sternly with, 'you have your orders, Lieutenant'. Deanna Troi is ranked Lieutenant Commander so he should have said Commander which is the shortened version of that rank, not Lieutenant.
When Q makes his first appearance on the bridge, he blocks two security officers from entering from the turbo lift, stopping them with an electronic screen and then shutting the doors. A few moments later when Q leaves, Tasha Yar comes forward to protest and the male and female security officers are suddenly behind her. The turbo lift doors were in the shot and never opened to admit the pair.
In the holodeck when Ryker crosses the stream he steps onto a loose rock it easily rocks side to side. When Wesley crosses the stream we see him fall from that same loose rock. Data jumps onto then stands on the same rock to lift Wesley out of the water but the rock is no longer loose.
Captain Picard is explaining to Data the details of the mission, Cut to Data sitting at the conn pressing buttons, paying attention to that. Cut to bridge fore and Data is looking up with a perplexed look on his face.
When the saucer separates, it is shown drifting away, on the view screen it goes a pretty far distance, yet when it changes screens, the saucer is closer to the ship then the view screen image. However, such effects are normal when two images are seen through different optical systems. For example, if the screen was set to a wide field display, then things could appear much further away than they really were.
When buying the fabric, Dr. Crusher tells the vendor to "charge my account" for the cloth. However, it is repeatedly stated there is "no money" in the Federation. How could you charge someone when they have no money?
Picard orders no transmissions or use of intercom in case Q is monitoring voice messages. He then orders Worf to send a text message to engineering. If Q has access to voice messages he could just as easily have access to text messages as well.
In the Holodeck, Data tells Riker that he graduated Starfleet Academy with the "Class of '78". A later episode, Star Trek: The Next Generation: The Neutral Zone, establishes that Season One is set in the year 2364. It is likely that this early in the season, the 2364 time line had not yet been established by the show's producers, otherwise this would create a plot hole that would imply that Data had graduated with the Class of 2278 and served in Starfleet for 86 years.