How far into Washington, DC should I drive before taking the Metro?

The commuter lots fill up very early in the morning on work days, and the fill up earlier and earlier as you look at metro stops closer to the city. The closest metro stops don't even have nearby parking. I'd pick one of the end-of-the-line metro stops and arrive early, maybe 8 in the morning.

I remember my first visit to DC where I tried to show up at a leisurely 9 or 10 and there were no spaces at Falls Church. Now a resident of the area, if I was going to do that again I would have tried parking at the Vienna-Fairfax stop.

Traffic on 4th of July is particularly bad even though it is on a weekend. When there is a rally or big even going on, sometimes not only is a tricky to get a parking place in the closer stations, you might have to take the train out to get a spot on a less crowded train going in. This was the case when John Stewart was in town.

I stay in the district, and if you were to ask any local their opinion about the metro, you'd probably have to cover your children's ears :). With that said, I would recommend riding the metro on special occasions, such as the July 4th fireworks display. For parking, come down (South) Wisconsin Avenue (exit 34 off I-495). Once on Wisconsin Avenue, you will drive across multiple metro stops - starting with the Medical Center, Bethesda, Friendship Heights and AU/Tenleytown. All of these are predominantly residential, with lots of street parking around. And considering these are uptown, away from the craziness, you won't be as overwhelmed with tourists rushing the trains, PLUS the ride will be shorter than the far flung stations. A word about street parking though - the district enforces the meters on Saturdays. Good luck, and enjoy your stay!

EDIT: Just noticed your final destination being on the Green Line - for that I would recommend parking in the Greenbelt lot - I believe that's commuter-only exit 25(?) off the I-95/495.

I lived out by Dulles Airport for about 4 years. If you know when to go, and some alternative routes, you can make it into the city and find parking with relative ease. That being said, 4th of July weekend is unique, and everything goes off the rails. For starters, most of the mall and surrounding streets are closed for the fireworks and events, and the metro will be extremely crowded, especially after the fireworks (see https://www.nps.gov/subjects/nationalmall4th/closures.htm).

Also, because it's a holiday, station parking is a bit different. I can't speak much about the red and green line (most of my knowledge is NoVA Orange and now Silver), but others are correct in that you should not count on overnight parking at the station lots. However, there is usually some parking around the station via street parking or 3rd party parking garage.

Most importantly, if there is no traffic, the metro is slower or similar to driving. Don't just stop at the head of the line to potentially avoid traffic. For example, Wiehle-Reston East to Metro Center via Metro is 45 - 65 minutes. Driving is 30 - 240 minutes. So the metro is a sure thing, but driving can be faster, or it can be way worse. Before the silver line came around, we would drive as far as East Falls Church and take the Orange line the rest of the way in.

Special 4th of July caveat As I said, it's a zoo on the 4th. It's going terrible from every angle. Arm yourself with knowledge of the possible routes, be aware of the parking along it and near your destination, and most importantly of all; in DC, sometimes it's faster to take an extremely convoluted route than straight in.

My advise, go as far in as you can until you hit traffic, and if the beltway has no traffic, get as far south as you can, because that's where you're headed anyway!