In my opinion the best way to travel around the island of mykonos is to book a private transfer!
You and your friends and family can care more about having a great time and less about bus schedules, transportation and so on.
I strongly recommend it as the costs are not too high. Medium budget holidays may be able to afford it!
The boat connection of Mykonos with Piraeus and Rafina ports is great. You can find many companies for Mykonos with regular ferries (about 30 euros one way, the ride takes 5,5 hours) or catamarans that goes faster (3,5 hours but double the price). You can also travel here by boat from other islands like Paros, Santorini etc
There are also daily flights from Athens with Olympic or Aegean airlines but also from other European cities and many charter flights too. The package groups come this way all the time. Mykonos airport is 3km out of Chora. Taxis will wait outside the arrival hall.
If you just want to see the main town (Chora) you will just walk around easily. The port is about 600m away from the center but if you arrive at the new port (cruises, old ferries) you will need a taxi as it is 3km away from town.
There are several buses that connect Mykonos town with the beaches (like Paradise, Super Paradise, Platis Yialos etc) many time each day till midnight. The bus station is at Hora, in fact don’t expect any bus station, you will see the buses parked all together next to a square :)
There are also some taxis but sometimes you will have to wait long enough for one to come.
Alternative you can rent a car or a bike and things will be better for you. There are several rental companies near the port
I met Mihalis Diamantis (Manager of Euro Club) many years ago when I was trying to find a cheap car for rent during my second journey to Mykonos.
Kind man, he solved my problem and was extremely flexible with my needs even I get some free days renting a car to them.
After several visits to the island now we are something very close to friends.
Each time I'm just arrived to Mykonos and he see me walking the narrow street right to his office (he's always reading on a coach on the street beside the entrance door) he smiles me and say "yassas Carlos" like if we were drinking a coffe the day before.
Euro Club is located downtown Hora (the port town) close to the Windmills area beside the parking street, and isn't a big company but a local company , anyway it was the best choice for me.
I know that renting a car in Mykonos is expensive, but the island worth a ride by her off beaten paths.
Rocky, rough, narrow roads cross the entire island offering nice views where tourists sometimes don't dare to ride. Don't miss, your money is well invested ;-D).
One of this companies is Europcar, correct services and cheapest in comparison with others agencies (except perhaps Euro Club), anyway, make your own scouting about because it can change.
Olympic Airlines is the main greek airline, so, they have several daily flights from Athens to Mykonos on high season, even from other islands (like Rodos for example) to Mykonos, with a wide fleet of commuters planes (hey, the small ones...).
Nice ride, over the blue Aegean sea at no more than 2.200 meters altitude.. Great experience.
The flight from Athens lasts more or less 40 minutes, a bit more expensive than other transportation ways but when you think about just 40 minutes...