Welcome to DC.
You have LOTS of nice scenic cycling options here.
I don’t know about the Sligo Creek Trail, but I can recommend the Capital Crescent, which is probably pretty easy for you to get to. It runs from Bethesda to Georgetown. Nice ride between cool destinations, and about as safe as any trail would be. You might get to Bethesda via the crushed stone part of the trail past Bethesda, officially called the Georgetown Branch trail. It might suck on a road bike though.
From there, you can go across the river to Arlington and connect easily to the Mt. Vernon Trail, which gives you a great river view and takes you down trough Old Town Alexandria to George Washington, and has two connections to the W&OD trail, which goes west for 45+ miles into farm country. Or you could stay on that side of the river and ride on the crushed gravel C&O Canal towpath, which goes 185 miles upriver.
Rock Creek Park also has a trail, and on weekends Beach Drive, which runs through the upper section of the park, is closed to motor vehicles.
Here’s a resource on the trails and routes:
http://www.bikewashington.org/trails
In general, all these trails are pretty safe using normal precautions.
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