Honestly, I didn't think that repeated use of "hope" didn't bother me because the paragraph flowed well, but here are some suggested variations if you want to change it up a bit:
By immersing myself in other cultures, listening to each individual’s stories and learning other languages, I seek {more active than just "hoping"} to have a more open mind, a kinder heart and broaden my horizons.
I hope to be more confident, more persevering and more inspiring. => I strive for greater confidence, perserverance and inspiration. {more determined, less wishy-washy}
And I resolve {more of a promise} to encourage the people I meet to achieve their goals and pursue their dreams, and to develop cooperation so we can all share a better world.