If it were possible to travel a million light years per minute, which
galaxies would one be able to arrive at within a very short time from
the moment of leaving the Milky Way Galaxy?  And in what order
would one arrive at these galaxies?

Within our understanding of physics, the question can't be answered
because you are postulating an impossible speed. Formally the
equations of relativity predict that if you go faster than light,
clocks will tick in imaginary numbers, which has no physical reality.
However, if you want to get to the nearest galaxy very quickly, you
only need to go very close to the speed of light. Then the Lorentz
contraction will reduce the distance to the galaxies. The faster you
go, the shorter the distance will be. Of course, it will still take
many years to get up to speed, if you want to avoid extreme
accelerations.

If you are asking which are the closest galaxies, check out this
beautiful picture of the current record holder for closest galaxy:

http://antwrp.gsfc.nasa.gov/apod/ap031117.html

The closest galaxies are, in order:

Canis Major Dwarf Galaxy (25k ly)
Sagittarius Dwarf Galaxy
Large Magellanic Cloud
Small Magellanic Cloud
NGC6822
Andromeda (2.2M ly)