Reagan Airport? Oh, you mean National. Locals still and always will call it National while tourist and newbies call it Reagan - so if you say Reagan expect a moment or two for it to sink in.
I would suggest taking a look at the hotels in
National Harbor. While it's not in DC it's close to National. It's new, has great hotels, restaurants and things to do in the evening safely by walking (I love Bobby McKeys, a dueling piano bar there). And it's not bad for being centrally located to not only visit the typical DC sites but also Virginia and Maryland. I'm pretty sure that Dave (Da Klugs) stayed there on one of his visits from what I remember.
In Virginia you will find Old Town Alexandria, Mount Vernon (George Washington's Home), a heavy dose of Civil War sites and battlefields, the Smithsonian Air and Space Museum at Dulles Airport and Arlington National Cemetery (The changing of the guard is something to see at least once) to name a few.
October would be a great time to visit. You miss the summer tourist crowds, the heat and stifling humidity. The trees will be changing colors, Oktoberfest will be popping up and around. And I'm pretty sure that the Maryland Renaissance Fair will be going on during your visit dates. Temps should be anywhere from the low 60's to upper 70's.
Transportation. I always suggest a mixture of rental car and public. The subways run through DC and into Maryland and Virginia. And you have the bus system. The metro system does not run 24/7. Keep in mind that millions work in and around DC. Rush hours here are extended both in the morning and evening. Taking a rental to a metro subway/bus location gives good flexibility.
Anyway - you have plenty of time to plan and explore all the options and sites.
Ron