When Lucy and the gang were in Europe, were they actually traveling or were they in a studio the whole time?

It was all in the studio.

I do find their background scenes were pretty good. I always marvel at the Lucy missing the ship scene. She runs up the plank to reach the boat. It looked like a real ship moving . A lot of prep work was done in that season

All ILL eps were filmed with 3 camera in the studio, right? There are occasionally a few scenes shot outdoors apparently for reality's sake, such the ship in the harbor and the Statue of Liberty, or a few of the driving scenes, like crossing the George Washington Bridge, or coming into Hollywood. Sometimes you can even make out what the 'actors' are wearing or their size, but I doubt if any of those scenes use the real stars.

Most of the studio props are good enough for a decent stage play, with the occasional exception of size perspective, lack of crowds, and movement. When they show a street/sidewalk scene in New York, there is always a noticeable lack of pedestrians, and the viewer is supposed to assume we are looking straight through the traffic that would be in the foreground. That's okay, but the lack of noise from that IMO is what makes it seem unreal. That is passable representation of the Constitution beginning to move, but there would be more people and probably a lot of confetti flying. And I doubt if the gangplank would be retracted if someone [Lucy] was still on it; seems there would be a security person on the dock end as well as on the ship to keep people away from the hatch.

One glaring oddity is in "The First Stop," where Lucy drives and gets suckered into making a long loop back to that ramshackle "Cabins and Cafe" place. The backdrop has a full moon, and it's in the same place when they stop to eat, when they leave the first time, and then again when they try to "sneak out." It should have disappeared from view. Also, I don't know why, earlier in this ep, they had to show an outdoor shot of 'Lucy' getting out of the car to look at 'Aunt Sally's Pecan Praulines' shop to see that it said "Out of business." But that shot is just in close enough to tell that's not really Lucille Ball.