Your best option is to drive! I recommend leaving early in the morning to beat traffic- there is a one lane highway and small bridge you will need to cross that can get insanely busy. If you are traveling from Long Island or want a more scenic route- there are Ferries you can take to C.T. to cut out apron 2 hours of your drive. It will not save time but you do get to stretch your legs and have a beer on the boat. Other options are to take the train/flight to Boston and catch a bus to Hyannis Port or rent a car. This will probably take the same amount of time once you deal with security and rentals.
There are many car rentals you can do straight from Boston Airport OR You can take a shuttle bus to Hyannis from Boston (https://peterpanbus.com/routes/boston-to-hyannis-bus/) and either call an Uber or rent a car from Enterprise which is nearby.
Yes, there is an airport in Hyannis. It is very small and generally expensive , Hyannis is generally a 30-45 minute drive to Harwich Port depending on traffic. Website for the Hyannis Airport: https://flyhya.com