Getting to Nha Trang is very easy as it is served by all the main transport routes.

Road
Nha Trang is roughly 6 hours from Mui Ne (Phan Thiet) and 12 hours from Ho Chi Minh City (Saigon). Buses are frequent and if you take one of the tourist companies' routes, comfortable, with AC and reclining seats. A one way journey is about $10, or cheaper as part of an open tour ticket.

It is also possible to get buses to Dalat (~8hrs), Buon Ma Thuot (~6hrs), and on up to Quy Nhon and Hoi An.

Train