Translation of rapidly in Spanish

We would love to have you here as our guests! My niece Katja had also many diffculties with the new surroundings. She is really upset with this new change. How was your trip to Italy? I'll keep writing to you!
The ride took us through mountainous areas, covered with dark green forest, with so many curves that I got sick. I wanted to vomit but I held myself back, as there were so many other next to me.
I boarded the AA flight to Cancun, and got there around two o'clock. It was my first time in Mexico. On the same night I had a flight booked to Havana, but I did not know whether I should take it.