It's still winter above 4,000-5,000 feet in the Pacific NW, and will probably not be significantly melted out until mid to late June (depending on amount of snow pack and temperatures between now and then). Even if you're experienced and have the gear for snow navigation, there is still often avalanche danger in the higher mountains through May. That's also true in the Rockies, although the altitudes are higher..
There are lower-altitude areas, such as some of the river valleys in the Olympic Mountains. There are also coastal trips in Olympic NP and California's Lost Coast, but I hesitate to recommend those to beginner hikers, because they are pretty rugged (muddy headlands to be scrambled over) and need careful attention to weather and tides.
You don't say where you live, so I don't know if any of these (in Oregon or nearby) would be worth the trip for you if you have to travel far. Winter and spring hikes
Just a few notes on a couple of these: Eagle Creek trail in the Columbia River Gorge is closed this year. The Wenaha River Canyon burned last summer, so contact the Umatilla National Forest on its status. Like the coastal trails, Hells Canyon is really rugged and definitely not for beginners.Edit, later:
Enchanted Valley in Olympic National Park is a mess this year--the access road is washed out, adding 6 miles each way to the hike, and portions of the trail are washed out and in really bad shape. However, the Hoh River valley might still be a good choice. The Olympic NP website (nps.gov) has lots of detailed trail descriptions.
Note that 2015 was a very low snow year, so many hikes were accessible weeks or months earlier than normal. Please don't plan based on May 2015 trip reports!
More: If you fill out your profile, including in which state or region you live, you may get a lot more suggestions for May backpacking rrom our members, who live all over the US, some in Canada and as far away as Australia. Regardless of where you live, somebody here will be relatively close!