What is the best Greek Island for a family 1 week vacation...able to
go on a day trip by boat to visist other Islands.
Whay is the "finest" hotel in the Greek Islands....regardless of cost

I would respectfully suggest that you find out a bit more about the
Greek islands from a guide book; refine your thoughts; and then
possibly redraft and repost your question. The point is that you can't
just lump them together. There are many dozens of Greek islands; but
they differ in size, remoteness, terrain, architectural and cultural
interest, availability of accommodation and transport, etc, etc. An
island that may be perfect for one family may be purgatory for
another. Your question is a little like asking what is the best US
city for a family holiday.

I can vouch for Rhodes, Skiathos and Cyprus.

These are three of the biggest and they are all GREAT!

I've been to some of the smaller ones as well, including Delos and
Mykanos but there's more to do on the bigger islands.

Hotels? I've stayed in lots of the 'finest' all over the world
(including the Presidential Suite on one occasion) and they are all
much of a muchness, except for the price tags.

Don't worry there'll be plenty of things to spend your money on ...

If you take your Good Lady with you.

The only link I have bookmarked is for Santorini as it looked like the
ultimate 'unwind' holiday.  Don't know how it would play for the
kiddos.

http://www.greekhotel.com/cyclades/santorin/oia/

I can also vouch for Cyprus as I lived there for 2 years.......but but
but I dont think the Greek or Turkish Cypriots would take kindly to
Cyprus being referred to as a Greek Island. I do suppose some tourist
books though may say that.... but the clever probonopublico doing it?
makes me feel bad.

Hi fredg2004-ga, I bow to your wisdom.

It's all Greek to me.

I agree with purchasing a Guide book or some DVDs because each island
has its own charachter speciality. Since you only have a week and are
taking the whole family I would not go to Mykonos because it is a
party island and depending on your sensibilities you may not want to
expose your kids to some of what goes on there.  Santorini is
beautiful and pretty well balanced but it may be difficult to travel
to and fro some of the greek islands especially with only 1 week. 
Rhodos is an awesome island because it has a little of everything and
is big compared to other islands.

Take a look at a book and DVDs and I promise that wherever you go you
will not be disappointed.  Hope my opinion helps a bit.