Hello! The correct answer is the last sentence.
“On Monday, June 21 I will be flying to the island of Hawaii. I will be spending a week in the Hilton Resort and will be visiting Dukes for a delicious meal.”
This is because of many reasons! To begin with, the first letter in a sentence is always capitalized. Additionally, months are also capitalized. Hawaii, Hilton Resort, and Dukes are all proper names so those are capitalized as well.
Hope this helps!