I think: I'll be interested to see what happens sounds more usual than I'll be interested in seeing what happens I'm interested to find out what happens at the end of the book. I'm interested in finding out what happens at the end of the book. Discover Learn Know These are all possible.

Sometimes they are interchangeable as in I like skiing and I like to ski. Sometimes only the infinite will do as in Hamlet “To be or not to be, that is the question.”

I think only certain verbs work with the infinitive, such as see, hear, learn, find out, discover etc. Those words also work the "in -ing" possibility. Another possibility is "It is interesting to ..." I'm interested in going to Russia. (focus is on me) It would be interesting to go to Russia. (focus is more on the activity, but the fact that I'm speaking means that the interest is mine) I think it would be interesting to go to Russia. (More explicitly my interest) But not "I'm interested to go to Russia."