Most people will enter the North Shore from Vancouver by road or through the Translink system.
Lions Gate Bridge
The Lions Gate Bridge is a Vancouver landmark linking the city with the North Shore across Burrard Inlet. Similar in style to San Francisco's Golden Gate Bridge, it was built in 1937-8 in large part to help develop land owned by the Guinness family of beer fame in West Vancouver. Although the family sold their land in the 50s, a link remains in the decorative lighting they donated for the bridge in 1986.
The Lions Gate Bridge
The bridge itself is three lanes with sidewalks on either side for walking or cycling. The center lane is controlled by traffic signals, so it alternates direction depending on the traffic conditions. The bridge is best avoided during rush hour if you're in a car, at least but otherwise provides a more scenic entrance or exit to the North Shore than the larger Second Narrows Bridge.
By ship
TransLink also provides a ferry option to get to the North Shore, called the Seabus. This is a passenger only ferry that goes from Waterfront Station in Downtown Vancouver to Lonsdale Quay in North Vancouver. A ticket or transfer used on a bus or the Skytrain is also accepted on the Seabus.
The SeaBus runs every 15 minutes until 6:45PM M-F and 10AM - 6:15PM on Saturdays. It runs every half hour at all other times. The SeaBus operates from about 6AM to 1:20AM, with shorter hours on Sunday. A schedule is available on the TransLink's website.
BC Ferries (http://www.bcferries.bc.ca) has a ferry terminal in Horseshoe Bay West Vancouver with three routes servicing it, including one from Nanaimo on Vancouver Island. The ferry terminal has a road that connects directly to Highway 1 and Translink bus connections as well routes 250, 257, 259 and C12.
By bus
The TransLink (http://www.translink.bc.ca) bus system connects both North and West Vancouver with the rest of the TransLink system in Vancouver and its surrounding suburbs. Going to the North Shore means crossing a fare zone boundary so a ticket to/from Vancouver, Burnaby or Richmond will cost $3.75 and a ticket to/from Surrey will cost $5.
By car
If you're coming from Vancouver, you enter the North Shore by either Highway 1 Trans Canada Highway across the Second Narrows Bridge into North Vancouver or by Highway 99 across the Lion's Gate Bridge into West Vancouver. Visitors travelling south on Highway 99 from Squamish or Whistler will arrive in the North Shore just above Horseshoe Bay.