When we first see the shuttle carrying Tom Paris, the registry number is 71325. In the second shot of the same shuttle, the registry number is 1701-D, indicating it belongs to the Enterprise. In the final shot, when the shuttle is flying over the saucer section of Voyager, it has returned to 71325.
After Torres & Kim are rescued on Ocampa, they are beamed directly to Voyager's sick bay, still wearing the clothes given to them by the Ocampan doctors. But when the scene segues back to Voyager's sick bay, Torres is wearing leather civilian clothes and Kim is wearing a uniform. (These clothes had been removed from them on the Caretaker's array.)
When Captain Janeway moves from the corridor into engineering, a tricorder suddenly appears strapped to her waist.
When B'Elanna Torres is restrained after waking up and trying to escape the Ocampa medical facility, the top button of her medical gown comes undone during the struggle, briefly exposing part of her chest, but in the following shot a few seconds later, her gown is buttoned up again.
Janeway's hair is unkempt and falling out of its pins after the ship is turbulently pulled into the Delta quadrant, up until the point she leaves the bridge for engineering--when she arrives in engineering it is up again, flawless to a degree no woman could manage herself in a turbolift without a comb or mirror.
Lieutenant Tuvok wears the insignia of a lieutenant commander throughout most of season one in an apparent costuming error. It will not be until Revulsion (1997) that he is actually promoted to lieutenant commander.
When Janeway first enters Engineering to investigate the warp core fracture, a technician tells her the pressure is 2100 KPa; she reacts as though that is dangerously high. Following the procedure to lower the reaction rate (and thus core pressure), the quoted pressure is 2500 KPa -- higher than before.
Nelix shoots water tanks with phaser. Should have blasted hole in tank, but people are getting water from an obvious faucet on tank.
Neelix is seen taking a bath in a tub. His foot emerges from the water and it appears human-like. In future episodes, it is "alien".
The scale of Voyager in comparison to Deep Space Nine varies between shots. At its largest, it is scaled larger than a Galaxy-class ship in comparison to the station. A Galaxy-class ship is about twice the length of the Intrepid-class Voyager.
The scale of the shuttle craft that brings Tom Paris to Deep Space Nine varies wildly between shots. In most cases, it is scaled up too large in comparison to the ship and the station. It would have to be several decks tall and would not fit into Voyager's shuttle bay.
Kes warns the others not to touch the energy barrier that is partially blocking the tunnel, as they have been warned it will "burn your skin off." Yet both she and Torres are seen passing their knees through where the barrier should cover without injury. The special effect for the energy barrier has been applied to the shot without regard for the actors' movements.
When the Maquis beam over to Voyager's bridge, they materialize using the new transporter effect developed for the Intrepid class ship. In Star Trek, different races typically have their own beaming effect, and it changes over time. Even if the Maquis ship was utilizing Federation technology, the ship is stated to be at least 39 years old and would not be using the newest, most advanced transporter technology that just went into service in Starfleet vessels.
When Neelix shoots the water containers with his phaser, as the Kazon are running to fill their buckets, you can see the water spigot with the water coming out on the right container.
Harry Kim is an ensign, fresh out of the Academy, and yet he is assigned as the Operations Officer of the ship, a member of the senior staff. With many higher ranked officers aboard Voyager, and many more to choose from throughout Starfleet, it makes no sense that an untested officer would be given such a position. For comparison, the Operations Officer aboard the Enterprise D was Data, who held the rank of lieutenant commander and had nearly 25 years of service before attaining that position.
Janeway uses tri-cobalt-equipped torpedoes to destroy the Caretaker's array, rather than allowing the Kazon to gain control of it. The array was their only way back to the Alpha Quadrant. With how advanced Starfleet technology is, it would be a simple matter to rig the torpedoes with a timer, set them to destroy the array, then activate the array to send them home before it is destroyed.
The Ocampa are so set on stopping any of their people from leaving the city by way of the tunnels that brought them there originally that they have erected energy barriers that will purportedly burn the skin off anyone that touches them. Yet they left the stairways bolted to the cliff-side intact.
The Ocampa elder tells Kim and Torres that he believes they were separated from their crews by the Caretaker to protect their crews from the plague they have been infected with. If they are believed to be carrying a deadly, infectious illness, they should be quarantined, not allowed to roam freely about the Ocampa city.
Speaking to Katherine Janeway, Tom Paris describes himself as "a mercenary, willing to work for anyone who'd pay my bar bills." The Federation does not use money. As long as he went to bars within Federation space, he would have no bar bills.
Tuvok makes an argument that any action the Voyager crew takes would affect the balance of power in that sector of space, and that "the Prime Directive would seem to apply." The Prime Directive is non-interference with the development of pre-warp civilizations. As the Kazon and the Caretaker were both post-warp societies/entities, and the Caretaker's actions ensured that Voyager was already involved in that sector's events, the Prime Directive would not apply.
Ensign Harry Kim is a newly commissioned Starfleet Ensign. The Star Trek Franchise has uses Navy / Nautical ranks, traditions and terminology. Ensigns are promoted without promotion boards in 2 years to Lieutenant junior grade. From Lieutenant junior grade, it is a minimum 2 years to full Lieutenant. Harry Kim must be the longest serving Ensign in history. Captain Janeway has the authority to promote all personnel under her command as she has the power to give "battlefield commissions" to personnel as she has done with Chakotay and Torres. The ranks of all of Voyager's officers and time in grade are questionable. Even prior to Voyager being stranded in the Delta Quadrant, senior officers and department heads on a ship of Voyagers size are junior officers and not appropriate for their positions.
Tom calls Voyager to beam up everyone except him. The crew man that responds to Tom calls him Sir. Tom was an observer at this point, and had not yet been given the rank of lieutenant.
Tuvok surmises that the Caretaker is dying. One of his pieces of evidence is that the Caretaker would not abandon his duty to look out for the Ocampa "after a millennium of providing for these people." The Caretaker never told the Voyager crew how long he'd been watching over the Ocampa. The Ocampa elder told Torres and Kim that the Caretaker had been providing for them for 500 generations. Given the lifespan of the Ocampa, 500 generations would likely be somewhere between 2000-4500 years, not a single millennium (1000 years). Tuvok seems to pull this number out of thin air.
Upon leaving the captain's ready room, Harry Kim looks around the bridge in amazement. He would have already had to pass through the bridge to get to the ready room.