Why Moria?

[Tuor of Gondolin]

I tend to agree with above arguments for heading East
or West, as opposed to South (although if the
Redhorn gate was open it would also be a good option).
They could have headed east past the Beornings and
then gone down the River Running, getting to the area
of the Sea of Rhun as a jumping off place for a dash to the
south side of the Ered Lithui (hey, beats passing by Shelob
and/or Minas Morgul as an entrance area!

Alternately, despite Gandalf's worries, immediately
after the Council of Rivendell head west to the Grey Havens,
get Cirdan to send a small fleet, 2-3 boats at most, south
of the Angren (any further by sea and there's corsair problems),
and then go by land east to Lebennin and South Ithilien.
Then either see the giant spider or head east around the
mountains and so into south Mordor.
Sea travel by elf sailors (hugging the coast) would be much
quicker, and being near shore they could evade any enemy ships
or stormy weather.

There are problems with any route. Difficulties with the sea
route include evading armies moving north, the length of the
trip around mountains (unless JRRT invents another pass- or
moves Shelob around ) and being stopped by Denethor.