@curly Sticking to bike paths and protected bike lanes makes things lots less stressful. Residential roads (like those on Capitol Hill) are also pretty calm. I usually try to avoid avenues without bike infrastructure (i.e. Rhode Island Ave or North Cap). Helpful bike map at https://godcgo.com/wp-content/uploads/DC-Bike-Map.pdf
@WABA offers group rides which might be nice (I haven't used them, though), and @hillfamilybiking has slow routes that I enjoy.
(I would worry less about rules--drivers don't really follow them.)
@curly What's the thing that you're finding uncomfortable while on the streets? Is it the streets you're on? Is it the cars? Is it not knowing the "best" routing? Is it generally feeling like you're just doing the wrong thing? (You almost certainly aren't!)
Lots of potential sources that I think riding more (and on protected facilities) would help with, but that might not be the right answer for some sources of discomfort.
@curly The rest definitely comes with practice!
The question of when to bike on the sidewalk mostly comes down to personal comfort and how many people are on the sidewalk. If you think the cars are too fast and close? Sidewalk it is! I often see bikers on sidewalks and think, "I'd do the same."
If you're deferential to pedestrians and there are no protected facilities nearby, it is ok to make sure you are safe. You got one body--take care of it!
@curly
(Are there rules about where you're "allowed to" bike on the sidewalk? Yeah. Fortunately, there's enough protected bike infrastructure downtown that going a few blocks out of my way to use it isn't a big deal.
But am I going to illegally use the sidewalks along Independence Avenue between 1st St SW and Washington Avenue SW nearly every day? Yes, and I ain't going to feel shame about it.)