Below site has COMPREHENSIVE information and timetables about Mumbi local trains:
http://mumbainews.wordpress.com/2006/08/30/local-trains-time-table/
Look for the direct Vashi Andheri Train times...
An amazing site that has more info than any official site for the same!
Buses
There's BEST buses connecting Vashi / Navi Mumbai to Mumbai and NMMT withing Navi Mumbai.
See this link for BEST bus routes using a cool software from IIT:
http://www.cse.iitb.ac.in/navigator2_gis/Main
