Please advise which is the best Greek island to visit in early April? What’s the best way to get there from Athens and how please.
Please define "best". What do you hope to experience?
Your travel options are ferries and planes, depending on the island. Some have airports, some don't.
I would look to the larger islands or the most popular.
Naxos and Crete both have large permanent populations, though Crete is more mainland feel than island due to the size, but Chania would still be nice in April.
Santorini has become a year round venue due to popularity, so you could spend time there.
Flying would be the best option, ferries might be a bit infrequent yet and subject to rough seas. Ferry to Santorini or Crete is very long, to Naxos it would not be terrible.
Early April 2018 (Easter) we spent a couple of nights on Hydra. They were certainly open. We only stayed for that short time because our main focus on the trip was driving and seeing the Peloponnese peninsula
We’d scheduled a ferry to Hydra from the Pireas port just south of Athens, but the seas were so rough that our first ferry couldn’t sail. The second one, an hour later, also got canceled, so we and another couple split the cost of a taxi to get to the Metochi port, some ways away, but with a shorter ferry crossing and calmer waters.
Not sure what you mean by "the best Greek island". I assume that you mean one with fairly stable weather conditions that time of year? I'd go down as far south as possible for that. Don't expect pleasant water temperatures for swimming though.
The seas can be rough any time of year in the Aegean. I have been terribly seasick on trips during summer even, I have seen the "Flying Dolphin" being cancelled and the Catamaran that was waiting at the mooring shaking so badly that I couldn't have got on if I tried. I have also seen planes being canceled due to weather conditions, especially the smaller ones going to the islands. So really, there's no guarantee of either.