To expose someone or something to a thing is to put the person or that something in a position where the thing can influence them or affect them in some way.
Do not expose the biscuits to air or they will go stale.
When paint is exposed to the sun the color fades.
A child exposed to music at an early age may become a musician.
We you travel to a foreign land, you are exposed to a new culture.
"ex" means "out" and "pose" is related to "position". The meaning is to put something out (where it can be affected in some way).
This locution is a passive construction. The transitive verb is expose.
Expose the wound to air so it may heal better. active voice
The wound was exposed to air so it would heal better.passive voice
Therefore, to find a substitute for "be exposed to" it must also have this passive meaning. Being exposed to a thing is something that happens to the thing which has been exposed.
You might say
We were shown another culture.
I had never been shown another culture until I went overseas.
When you open yourself to other cultures, that is something you are doing on your own behalf, so it is not a perfect substitute for "be exposed to", yet it has a similar meaning of allowing yourself to be influenced.