When Bill and Ted tell Socrates in his original time, 410 B.C., that they're dust in the wind, Socrates says, "Yes! Like the sands of the hourglass!" Although the joke is that he's unknowingly quoting the opening of Days of Our Lives, the hourglass was first introduced in Europe in the 8th century A.D.
When the dudes pick up Socrates in 410 BC, the buildings of Athens are stark white. Although surviving ancient Greek buildings are known for this complexion, in their heyday they were painted in bright multicolor.
A 15th-century commander speaks of executing prisoners in the Iron Maiden - which, as the joke, is misunderstood by Bill and Ted to mean Iron Maiden. This instrument of torture was invented in 1793. Ironically, it was built specifically for the purpose of being a decoration in a house-of-horrors carnival, rather than for actually harming anyone.
When Bill reads the assignment to Ted, he says, "Express to the class how an important historical figure from each of your time periods would view the world of San Dimas, 1988." His lips are actually saying, "San Dimas, 1987". (The "1988" was dubbed later because of a delay in the movie's release.)
During their report, Bill and Ted say that Joan of Arc drove the English out of France and had the Dauphin crowned by the time she was 17. Joan was 17 when she began her career and had the Dauphin crowned, but the English were not driven out until 1453, the year France won The Hundred Years' War.
When Bill and Ted pick up Genghis Khan in Outer Mongolia, it's the year 1209 as indicated on the screen. However, during the report, Bill says they picked him up in the year 1269; Genghis Khan died in 1227.
Napoleon looks at and tries ice cream as if he's never seen such a food. In reality, ice cream had been around for generations prior to Napoléon Bonaparte's lifetime. Catherine de Medici introduced the French to ice cream when she married the future King Henri II over 250 years prior to Napoleon's reign. Being a dessert of the elite, Napoleon more than likely had ice cream before his time travel to San Dimas.
In the saloon fight when Billy the Kid is caught cheating at cards, a prostitute swings in on a rope and takes a guy through a window with her. The rope she swings on disappears and re-appears a few times between shots.
After the "gum and cans" repair done to the booth's antenna early in the film, some later shots of the booth show an undamaged antenna. This is most noticeable in the "Circuits of History" as well as the presentation at the end of the film.
Beethoven is picked up by the dudes in 1810. By that year, the real Ludwig van Beethoven was almost completely deaf. Yet Beethoven responds to characters' spoken words, musical notes, and other aural cues at several points during the film.
During the water slide sequence, Napoleon appears to be wearing modern underwear underneath his historical undergarments. He could have somehow acquired these undergarments during his time in 1988 San Dimas.
When we see the phone booth being formed and there is a reflection of it in Rufus' glasses, we can see that the "reflection" is just a photo of the booth as the text at the top of the booth is NOT reversed.
The goof items below may give away important plot points.
When Bill and Ted are at the Circle K and the phone booth lands with their future selves in it, you hear Bill say "Aw, I hate that part" just as they land in the booth. When we see them land at the Circle K to talk to their past selves at the end, Bill doesn't say that.
After finishing their History Presentation and returning all the Historical Figures to their own time periods, Bill and Ted are back in the garage filming a rock video and the Time Machine is shown standing outside the garage door.
Minutes later there is a flash and Rufus arrives with the Princesses in his own Time Machine, but when the garage door is opened there is only one Phone Booth standing there when there should be two.
At the beginning of the presentation, Billy the Kid shoots out a light with his gun, despite having just been broken out of jail, where the police would have undoubtedly taken his gun. However, the presentation includes a couch for Freud Dude, swords for Miss of Arc, keyboards for Beethoven, a halberd for Genghis Khan, and battle maps/pieces for Napoleon, so it's safe to assume that future Bill and Ted got Billy a gun for the presentation when they came back to set everything up.