I'm a Washingtonian; you are going to love the DC metro area.

Unfortunately for you, neither Russia (nor any part of the former USSR) was party to the 1949 Geneva Convention on Road Traffic which would allow you to drive in the US for up to a year with a foreign driving licence.

You can drive with your IDP, accompanied by your Russian permit, but will need to go through the process of obtaining a State-issued permit. In both locations, however, the road skills test might be waived but the testing center would have to confirm that for you. While that may be so, I've left that requirement, as stated in the relevant sources, because there may be variables in your circumstances.

Here's what [Virginia requires][1], that within 60 days of moving to the State, you must obtain a Virginia driver's license. There is a fee, currently $32.

>If you are age 18 or older and you hold a valid driver's license issued by a country other than Canada, France, Germany, or South Korea, you may not be required to show proof of successful completion of driver education or hold a learner's permit for 60 days. However, you must pass the two-part [knowledge exam][2], road skills test and a vision screening.

The documentation you will need to provide to prove your

 - Identity (e.g. birth certificate, passport, government-issued ID card)
 - Legal presence in the U.S. (e.g. U.S. passport, permanent resident card, foreign passport and I-94 print out)
 - VA residency (e.g. rental agreement, recent bank statement, recent paycheck stub)
 - SSN, if you have one (e.g. Social Security card, paycheck stub, Social Security statement)
 - Name change, if applicable (e.g. marriage certificate, divorce decree, court order)

Here are the [VA DMV's list of required and accepted documents][3]. 

The [District of Columbia has two different licenses][4], the REAL ID driver license and the Limited Purpose ID driver license. The latter is for those who do not have a Social Security number (never have had one, won't be getting one).

The [documentation and requirements][5] are similar and, again, there is a fee. 

>You must also bring the documents that prove you are eligible for a driver license. The documents required are listed below, and DC DMV will not accept any documents that are not listed:
 - [Proof of identity and age][6] 
 - [Two documents that prove you reside][7] in the District
 - Proof of your Social Security number 
 - Vision Screening
 - [Knowledge Test][8]
 - Road Skills Test
Non-US citizens may obtain a DC DMV REAL ID driver license or DC DMV REAL ID identification card if they possess a USCIS Notice of Action, Form I-797, reflecting that they have applied for an employment authorization card, permanent resident card, or adjustment of status or any other unexpired document issued by USCIS or US Department of State allowing lawful presence.

  [1]: https://www.dmv.virginia.gov/drivers/#eligibility.asp
  [2]: https://www.dmv.virginia.gov/drivers/#know_exam.asp
  [3]: https://www.dmv.virginia.gov/webdoc/pdf/dmv141.pdf
  [4]: http://dmv.dc.gov/service/driver-licenses
  [5]: http://dmv.dc.gov/node/185742
  [6]: http://dmv.dc.gov/node/1115566
  [7]: http://dmv.dc.gov/node/1115502
  [8]: http://dmv.dc.gov/node/1119296