Why Moria?

[Gwaihir the Windlord]

Faster to go to the Gap of Rohan?

Such a journey may take a year or more.

A long trek through the deep winter wilderness, and through the semi-hostile Dunland, would not have been a good idea -- neither would it have been faster. Moria took about three days, and was far better in terms of secrecy; the Ring needed to be kept hidden and they had to go by stealth where it was possible. I believe Gandalf stated secrecy as one of the reasons for undertaking the Moria route? Either way, if you have a look at this part of FoTR it tells you, if I remember rightly, why Moria was the chosen.