I had the joy of living in the Washington, D.C. metro area for 18 months - you are going to have a BLAST.
The closest airport to Chevy Chase (MD) is the Ronald Regan National Airport. There's a metro station right at the airport. Check with your hotel and see if they have a shuttle from the airport to the hotel. Also, check the hotel web page for transportation options to / from the airport. The other option is Baltimore Washington International Airport (BWI). That airport also has transportation available - just check the website also.
In Bethesda - there is a metro stop also. Really, don't rent a car - USE the metro. The way it works is that there are five colors of rail lines (red, green, blue, yellow, orange). The trains are named for their end points. Read everything you can on www.wmata.com - it's the metro web site.
It's a HOT walking city - plan on water breaks; wear sneakers and sunscreen.