If you want to drive 66E all the way into the city, you'll have to plan ahead a little. First, don't expect the trip to be fast anytime before about 9:30 or 10:00 in the morning on a weekday, or anytime after about 3:00pm. Also, you must have at least two people in the car -- or have a Virginia clean fuel license plate issued before 2006 -- to drive on the segment of 66E past the Beltway (I 495). Once you get into the city, you'll have to find parking, and that will be both difficult and expensive.
Your best bet from Herndon is to drive to the Vienna Metro stop and take the Metro into the city. Parking at the Metro stop will cost about $4.50, which you have to pay with a SmarTrip card (a card costs $10 and comes with $5 credited on the card; you can use it to pay for Metro fares too). Visit
http://www.wmata.com/ to calculate the fares between Vienna and your destination.