North Vancouver is your best bet for family-friendly but not too expensive accommodation (especially Lynn Valley area); some parts of Burnaby will also be fine. Port Moody is MILES away from downtown and will probably mean a dreadful commute and will definitely mean you spend most of your time way out in the suburbs rather than enjoying things in and around Vancouver proper.
Furnished accommodation is not very common here and tends to be more expensive and more corporate - downtown flats etc. Try some property management companies - adventvancouver.com, vistarealty.net and see if they have anything to offer.
My experience as a SAHM in Vancouver is that many of us spend our time going around to the many, many fabulous family activities on offer, both outdoors and in. There's Grouse Mountain, Science World, the Aquarium, Granville Island, community centres everywhere, libraries, beaches, parks, pools, soft play, petting zoos, etc etc. It won't be hard to meet people if you sign a child or two up for a regular class/preschool at your local community centre. Good luck!
Oh, and before I forget: don't do the usual UK thing and assume that because it's Canada, you're going to have piles of snow every winter. It barely snows at all in Vancouver, and it's usually quite mild. Bring your wellies instead!