You can get from Mykonos to Santorini, and of course from Santorini to Mykonos, by ferry, by plane (sort-of) or even by helicopter.
Mykonos is about 150 km (93 miles) due north of Santorini and the two islands are connected by ferry, by domestic flights (though only via Athens) and - most exciting and most expensive of all - by helicopter.
The ferries between Mykonos and Santorini take 2-3 hours, sometimes even longer, depending which kind of ferry you take and the route they take. Some ferries are direct and some stop at other islands on the way, such as Paros, Naxos, and Ios. There's usually very little difference in the fares so if you're island-hopping on a budget you won't save anything by choosing to take a longer ferry ride.
Ferries between Mykonos and Santorini only operate in the holiday season, roughly from March to October. If you're travelling in either of those two months, double-check the details and don't assume there'll be a ferry every day.
Several different ferry companies operate services between Santorini and Mykonos, so it can pay to shop around for cheaper fares or times that are more convenient for you. Rather than go to each separate website, it's much easier to use an online timetable and booking service like Ferries in Greece that searches all the available ferry lines and ferry routes for you.
Mykonos and Santorini both have international airports and receive international flights during the holiday season. However, there are no direct flights between the two islands. The only way to do it would be to fly from Mykonos to Athens (25 minutes) and then from Athens to Santorini (45 minutes). That doesn't seem too bad but by the time you factor in check-in times and changing planes, it's much faster to take the ferry.
Only if you have a fear of ferries or refuse to travel by sea should you consider going from Mykonos to Santorini by plane. However, it's worth knowing about the options in case you need to get from Mykonos to Santorini in winter when ferries aren't operating. Olympic Air has daily flights between Athens and Mykonos, and between Athens and Santorini, all year round.
There's no doubt that the most spectacular way to get from Mykonos to Santorini is by helicopter. Imagine arriving on Santorini that way, especially if it's a first visit. It's definitely something to be regarded as a special occasion treat, though, for most people, like a honeymoon or big birthday. The price is several hundred euros, but despite that the flights do book up so don't wait till you're actually on Mykonos or Santorini to book. You might strike lucky, but don't bank on it.
To start your search for helicopter flights between Mykonos and Santorini visit the websites of Helistar, Icarus, The Greek AirTaxi Network, and HeliAlpha. Some companies also operate tours of each island by helicopter, so if you don't want to fly between them you can still see either of them by helicopter.