Learn English – Should I say “I am caught” or “I have been caught” in this situation

"I have been to" would always be correct, whether it's a city, a museum, or a friend's house.

"I was at" a museum would be correct, or "I was in" a city.

There is a bit of nuance here: If someone asked, "Where have you been all day?" one might say, "I've been at the Met. It was great!" However, if someone asked, "Have you ever been to the Metropolitan Museum?" or "Have you been to any good museums?" then one would say, "I've been to the Met. It was great!"

Side note: "at" is often used incorrectly by American English-speakers, e.g., "Where are you at?" or "Where's that at?" In both examples, the "at" should be omitted.

Buying is seen as an event rather than a continuous activity, so you can't use present perfect continuous in relation to buying a single house. You could use it if you were professionally involved in the purchase of houses, even though it's a number of separate activities over a long period of time:

I have been buying houses for 10 years

When you want to describe the purchase of your own house, you would use past simple to describe the purchase, and present perfect continuous to describe the payment by instalments:

I bought this house ten years ago and I have been paying for it ever since