It was long conventional wisdom to avoid driving and to take the Metro everywhere; unfortunately, the Metro has been very unreliable, particularly on weekends, and as a local resident I can no longer endorse it in good faith. To give you an idea, Metro's current public relations campaign is literally called Back 2 Good.
That doesn't make driving necessarily better, of course. Since Independence Day is on a Wednesday this year, I would not expect any significant change in traffic the weekend of the 9th compared to any other summer weekend. April through August is peak season for family trips to the capital.
So with all that preamble, I would point out that you can take a taxi or a service like Uber or Lyft to the Tenleytown-AU station on the Red Line, which about the same travel time as Friendship Heights but a stop closer in. This will cost a little more round trip than driving to Bethesda and parking in one of the county garages (there are no Metro-operated garages in this part of Montgomery County), but it's also a little less hassle. The public garages in Bethesda or Friendship Heights are unlikely to fill up on a weekend morning. You could also price out a ride from Brookmont all the way into the city, but as Brookmont is a fairly suburban area, the supply of drivers is not likely to be high, and the rates commensurately higher.
On the flip side, for a group of five, the aggregate round trip weekend Metro fare from Tenleytown to Smithsonian would be over $25, so driving together and finding a garage or lot downtown or in the Federal Center SW or L'Enfant Plaza areas is far from unreasonable. A simple web search for DC parking should turn up any number of websites and mobile apps where you can search and compare hours, prices, and locations. Some locations do accept reservations, but it isn't usually necessary on weekends— as noted, parking demand is mostly driven by commuters. By the same token, many garages are closed on weekends. Parking will be significantly more expensive on weekdays.
There is street parking along the Mall, but it is extremely limited relative to the number of visitors, and I would not waste time on a three-day trip hunting for it. There are a handful of metered lots in East Potomac Park near the Jefferson Memorial. Street parking downtown is metered weekdays and on Saturday, the price and hours and availability depending on the neighborhood.
As for Georgetown, I have not attempted to find street parking in many years. There are several garages in Georgetown, mostly in the complexes along the Potomac waterfront, or you can take Metrobus 30/31/32/33/34/35/36/37 from Friendship Heights down Wisconsin Avenue. This route has relatively frequent and reliable service, the ride taking about a half hour. For yet another option, there is a DC Circulator shuttle to Georgetown that runs from the Dupont Circle Metro, or it is a long walk from Dupont Circle, Foggy Bottom, or Rosslyn.
