Both Oregon and Washington state parks are hosting New Years Day 2014 guided hikes and snowshoes this year.
Oregon has 30 hikes to choose from, and this article from Terry Richard in The Oregonian lists them all. The closest ones to Portland are at Stub Stewart, Milo McIver, Tryon Creek, Historic Columbia River Highway State Trail, Rooster Rock, Champoeg, and Silver Falls.
Washington is getting in on the First Day Hike action, as well. Their press release lists 15 parks hosting a hike. Looks like the closest one to Portland is Goldendale Observatory State Park in Goldendale. Too bad Beacon Rock isn’t doing one!