Here's a query to return a list of all Mining trainers:
http://wow.allakhazam.com/search.html?q=Mining+Trainer
The closest to Darnassus will be Darkshore, your level 11-20 zone. Take the portal in Darnassus behind the bank and then talk to the gryphon master.
On the left is a link to List of Trainers. It's near the bottom of the navigation menu. There you can find the list of Engineering trainers.
Edit:
P.S. You are a Night Elf not a Dark Elf ;)
Edited, Dec 7th 2006 10:02am by thermalnoise