I still have a few doubts on the use of to/at/in after have been. Thank you for helping me.

1) I haven't gone out for a walk.
I have already been to school (or at school) today.
I haven't been to the cinema yet today.
2) I've already gone shopping online.
I've already done online shopping.
I've had this scooter for three months. (Not: I have gotten this scooter)
3) Katie, who likes rock music, has listened to it since she was twelve.

These look correct to me.

All are fine.

Sure! I'd be happy to help clarify the use of prepositions (to, at, in) after the phrase "have been."

1) "I haven't gone out for a walk."
In this sentence, "go out for a walk" refers to leaving your current location to take a walk outside. Since "go out" implies leaving a place, you would use "for a walk" to indicate the purpose or destination of your going out. So, you wouldn't typically use any preposition after "have been" in this case. The correct sentence is: "I haven't been for a walk."

"I have already been to (or at) school today."
In this case, "school" can be considered a specific destination or location. You can use either "to" or "at" depending on the context. If you want to emphasize that you physically went to the school building and attended classes, you would use "to." If you want to convey that you were present at school, but it doesn't necessarily mean physically being there (e.g., attending online classes), you can use "at." So, both sentences are correct: "I have already been to school today" or "I have already been at school today."

"I haven't been to the cinema yet today."
Similar to the previous sentence, "cinema" is a specific location. Here, you're expressing that you haven't visited the cinema up until now. So, you would use "to" after "have been." The correct sentence is: "I haven't been to the cinema yet today."

2) "I've already gone shopping online."
In this sentence, "gone shopping online" means you have already engaged in the activity of shopping through online means. Since you are not physically going somewhere, you don't need to use any preposition after "have been." The sentence is correct as it is.

"I've already done online shopping."
This sentence also conveys the same meaning as the previous one. Both "gone shopping online" and "done online shopping" mean the same thing. In this case, "done" is used to indicate that you have completed the activity of shopping online. You don't need to use any preposition after "have been." The sentence is correct.

"I've had this scooter for three months."
In this sentence, "had" indicates possession of the scooter. Since possession doesn't involve a physical movement or location, you don't need to use any preposition after "have been." The sentence is correct.

3) "Katie, who likes rock music, has listened to it since she was twelve."
In this sentence, "listened to" is an action that doesn't involve a specific location or movement. You don't need to use any preposition after "have been." The sentence is correct.