Ricky says "You couldn't even get through the Holland Tunnel" with the car packed like that. But as the cast sings "California Here We Come" the car is seen crossing the George Washington Bridge.
At the beginning of the episode Lucy tells Ethel of all the places the couples want to see while driving across the country. Lucy than pulls out a map of the country with a route drawn out marking the stops along the way. However, the route on the maps misses some of the sites that Lucy mentions. Case and point Lucy says Fred wants to see Niagara Falls yet the route drawn on the map goes nowhere near Niagara Falls.
When the Ricardos and the Mertzes are leaving their apartment building in the Ricardos' car (on their way to Hollywood), you can obviously see the tow wires towing the vehicle.
Lucy's mother sends a letter intended for her to the Mertz's because she claims to not being able to remember the Ricardo's address. However, the Mertz's live in the same building meaning their address is exactly the same as the Ricardo's. All Lucy's mother had to do was address the letter to them.