The Skytrain is the most efficient way of getting from New West to downtown Vancouver. There are five different Skytrain stations (I'm quoting from their website here):
22nd Street Station
On 7th Avenue at 22nd Street in New Westminster. Close to Marine Way and Queensborough.
At Columbia and 8th Street in New Westminster. Opposite the New Westminster Quay.
On Columbia at 4th Street in New Westminster.
At the foot of Keary Street in New Westminster.
Located near Braid Street and Brunette Avenue.
New West is easily accessable from downtown Vancouver, Burnaby and Surrey. You can get there by taking the skytrain eastbound and getting off at the New Westminster or Columbia skytrain stations. It takes approximately 20 minutes to go to New Westminster from downtown Vancouver by skytrain.
The skytrain is a light rail transit system (ie: monorail) - owned and run by Translink. I believe $2.50 buys you a pass onto the skytrain which also allows you access onto the bus anywhere in Greater Vancouver. More information, including the skytrain map, is available on the Translink website.