Since you are coming from the south, I assume if you would park at Metro it would be on the Blue or Yellow line
Map: http://www.wmata.com/rail/maps/map.cfm

Here is the scheduled track work for this weekend:
“Blue and Yellow Lines – 10 p.m. Friday through closing on Saturday
•Blue and Yellow Line trains will share one track between National Airport and Braddock Road as personnel conduct rail renewal work.
•Blue Line trains will depart endpoint terminals every 24 minutes.
•Yellow Line trains will depart endpoint terminals every 24 minutes operate between Huntington and Mt. Vernon Sq. stations.
•Blue and Yellow line customers should allow about 20 minutes of additional travel time.”
http://www.wmata.com/about_metro/news/PressReleaseDetail.cfm?ReleaseID=5061

Here is a parking garage related website:
http://washingtondc.bestparking.com/index.php

I would probably just drive all the way but that is because I find all the track work, etc. tiresome. I don’t have any advice about traffic.

On weekend days, traffic in DC should be lighter than traffic outside of the city which is congested as a daily fact of life.

Park at Union Station for $20 max and take the metro from there. Park at National Airport for $20 max covered or $12 max uncovered and take the metro from there. Parking for a flat $10-12 a day can be found at lots around town.

Parking at the zoo runs about $20. Best time to see the zoo is first thing in the morning so you should have little trouble getting a spot in the zoo's lot.

Parking at the National Mall. You can park on the street for free on weekends. Downside is it's limited to 2 hours and is hard to come by. Meter parking runs about $3 an hour, some take credit cards.

No, the protests should have no impact on your plans. You can reach either the National Mall or the Zoo without having to drive in the immediate vicinity of Freedom Plaza.

Actually, noting the above link, Sunday is the date rescheduled from Hurricane Irene for the opening of the Martin Luther King memorial. Driving downtown and travelling on Metro will be very on Sunday. The planned metro works won't be ongoing Sunday but will on Saturday.

*very heavy on Sunday

I would just drive directly into D.C. and find a parking garage on Saturday. (Parking is only free on Sundays now.)

While parking and riding Metro from outside D.C. could be a little cheaper than parking in a garage in D.C., it's going to be a whole lot more stressful and annoying with the track closures. I got caught up in that on a Saturday afternoon a couple weeks ago trying to get home on the Orange Line from D.C. I waited nearly 30 minutes for a train and the platform was jam-packed by the time the train arrived. (The train was jam-packed too.) I felt so sorry for the visitors who were trying to navigate Metro for the first time that day.

There is an Amtrak train that leaves from Richmond at 9:30AM on Saturday. It gets into Union Station about 12:30. Fare is $64 for 2 each way. Union Station is on top of a METRO station.

You can reserve a parking spot at the zoo for an extra fee. Visit the zoo web site for more info. Do it quickly, you have to reserve 48 hours in advance.

AMTRAK is certainly nice if you are 65 or over. On a weekday, the roundtrip tickets for two was about the same as parking in DC.

There are more trains from Staples Mills in Richmond than from the station in Shockoe.

Thanks to all. We decided to take Amtrak. With the senior discounts it cost a bit more than driving but not having to deal with traffic and parking made it worthwhile.
It's been a number of years since we drove up in that area and I'm sure the traffic hasn't improved any.
Thanks again.