Welcome to OB and Washington, like
@OTH Overland said a lot of stuff around the NW is about to close for the season but there is lower elevation place to go or head to the coast.
Another bit of info to help people out with suggesting trails would be what kind of rig are you driving and is it stock or modified?
If you haven't looked into them yet check out the Washington Green Dot Roads. It's a system of "trails" put together by BLM and Fish and Game. I say "trails" because all of them that I've been on are easy and most could be traveled in a stock rig, I haven't been on them all but I've been to most of the regions and that's what I've seen. They would be a good way to get off road and get your feet wet without getting in over your head. But with anything off road weather can change that pretty fast. Some of the roads are open year around and some are closed for snow or seasonal animal habitat. They're worth a look and most are within a couple hours of you. One nice thing about the site is the maps are downloadable and they have them available in a format for printing.
Want a great way to get into the Washington backcountry? Follow the green dots. Together with the state Department of Natural Resources (DNR) and private landowners, the Washington Department of Fish and Wildlife (WDFW) cooperatively manages many roads within Yakima and Kittitas counties under a...
wdfw.wa.gov
Another source for trails on this site is the Overland Bound Routes: Downloadable section. Go there and filter by NW region and a few will pop up, 2 that I've been on that others have posted are Little Bald Mtn. and Kendall/Skyline Rd. Both are easy drives and offer camping along the way, but I think they both might be getting snow by now. Little Bald Mtn. is in an area referred to as "Clover Springs" there is a lot of other forest roads to explore up there and places to camp all over the place. There is harder 4x4/Jeep trails up there but if check your maps and read the posted signs you can avoid them until you are ready. Kendall/Skyline is a nice drive along the tops of the Blue Mountains running from Dayton, WA down to Jubilee Lake at Tollgate, OR and you can go even farther south on forest roads if you really want. Again tons of camping along the way ranging from improved camp grounds (vault toilets and fire rings, no water) to primitive dispersed. Most of the route is graveled forest road but there is a section that is dirt and becomes muddy and rutted in fall/spring, closed in the winter. Best time to hit it is early summer before it's super dusty and fire season. Also with this route you can turn off mid way to go to Walla Walla if you want a shorter run or just feel the need for some wine tasting/ breweries. In the downloadable section there is also a route I posted titled Bethel ridge loop, it's made up of part of section 2 of the WABDR and its bypass put together in a loop. This is another run that might be getting snow already and the trail portion of that gets pretty muddy and rutted up when wet. When it's dry it's pretty easy, but I haven't been across it since a fire went through last year so not sure of its current condition. I don't recommend the trail portion for a beginner when it's wet, the rest is just forest roads that are super easy and offer nice views and camping. I do have tracks/routes from trips to these areas if you're interested, I just never posted trip reports on them.