There is one way that we can expose ourselves..

Both of these sound stilted to me. I can't recommend a better statement since I'm not sure what you are trying to say, but I would not use the expression "expose ourselves" as it has negative connotations in US English (as in exposing private parts of the body).

People can expose or reveal themselves through their writing.

They can expose their thoughts and desires, feelings and fears, likes and dislikes, etc. for others to see.

Not just US English.

Both are correct. It is very common to hear someone say, "Let me show you a way that can be done". I understand that you are viewing way as a synonym for path, and you walk through a path. But your friend's way is as correct as yours.

The other posters are right about the phrase "expose oneself", in that form it's generally referring to revealing one's private parts in public. Just to add, a very common joke about it is the fellow who stands on a street corner wearing just sneakers and a buttoned raincoat, even on a sunny day. And when someone passes by, he unbuttons the raincoat and opens it wide.

However, the word to expose by itself does not have those connotations. That's why large commercial fairs are frequently called "expositions". It is very common for a business executive promoting an item to say something like, "Through radio advertising we have gained good exposure in the Houston market", meaning that people know about his product.

Gaining exposure = getting to be known, most often in a good way.
Exposing yourself = not good.

I'm having a hard time sorting out what you're trying to express.

Is it that you are learning about new things or that you are demonstrating your knowledge about new things?

Neither of the examples you've offered precisely expresses either idea.

Definitely stay away from the exposing oneself phrase unless the rest of the sentence is very clear.

I should've have provided more context.
Thank you, ehBeth.

I'll keep in mind what you said.