Easiest route San Antonio to Wash DC?

I would work my way up to I-81 and run it up to I-66 in Va and then take it east into DC. That said, I don't know what you want to do on the way up there, but there are a couple of ways to go that way.

Go up to Dallas, take I-30 to Little Rock, AR, pick up I-40 and take it to Knoxville, TN where you get on I-81.

Or you can run I-10 to I-12 in LA, take it to pick up I-59, go north to Chattanooga, TN where you pick up I-75, (via a few miles of I-24 east) go north to Knoxville, get on I-40 east to I-81 north.

I don't care to run east on I-10 from Texas to New Orleans, but that is my personal choice.
Well, I am in a truck, so my choices to stop at are much more limited than yours may be, but I like to run up on the first way, but look on a map, if there are places you want to see on either route, take the one that suits your needs.

 

From San Antonio take I 35 to Dallas get on I 30 to Little Rock then run I 40 all the way to I 81 into VA then go to I 66

 

I used to live in San Antonio so I hope it helps