Continuity mistake: Trevor is driving the Ecto-1 and complains that Podcast is breathing on him. We see Podcast very close to him. The shot cuts and Podcast is now further away.
Continuity mistake: Phoebe and her mother are by the door of the farmhouse, waiting for the pursuing demons to appear. The shoulder strap of Callie's dress is lowered in most shots, but goes back up suddenly (and only) in an intermediate shot.
Other mistake: When Trevor attempts to start Ecto-1's engine the camera shows a very incorrect engine. A Triumph Herald/Spitfire 4 cylinder that is less than 1 to 1.5 litre.
Character mistake: There are signs out front as the Spenglers arrive at the dirt farm. One of the signs is a call back to a scene in Ghostbusters 1984. In the scene, Ray quotes the book of Revelation: "And I looked, and he opened the sixth seal, and lo, there was a great earthquake. And the sun turned black as sack cloth, and the moon turned blood." When Finn Wolfhard reads it he misquotes the line saying "The sun became as black as cloth."
Deliberate mistake: The mother jumped through the window, shattering the glass, but did not get cut or tear her clothes. (01:28:55 - 01:31:10)
Continuity mistake: When Gary is freed from his ghostly form he says "Oh, that was weird" and a shot later his left hand is suddenly next to his face.
Continuity mistake: After testing the proton pack, Phoebe raises her hand to adjust her glasses. A shot later her hand is lowered.
Other mistake: Egon's old jumpsuits fit the kids perfectly, but he was a grown man and these are young children.
Other mistake: When Phoebe calls Ray from the police station, Ray says the firehouse is now a Starbucks, but during the end credit scenes Winston is standing in the firehouse and Ecto 1 follows him.
Suggested correction: Ray was being sarcastic. A common joke in NY is that every time a business closes, the location becomes a Starbucks.
Suggested correction: This mistake has already been corrected. Starbucks is often used as a shorthand joke about general gentrification, companies going out of business, company over-exposure, etc. (Which is kinda based on reality. Ex. In my hometown there are literally four Starbucks locations over a 5-mile stretch of road, at least one of which I believe took over an old store that closed down. It's utterly insane.) Ray isn't being literal when he says the firehouse is a Starbucks - he's just making a sarcastic joke about how it's closed down.