Washington Monument is the only ticket I remember.Free of charge,but first come,first served.You can reserve online for the day you want,but do it ahead of time.Other places I don't think you'll need anything .Whitehouse you need your Congressman to set you up.Takes about a month ahead of time. We just spent two weeks there and could have spent two more.
If staying at Cherry Hill,it's a pretty good place.Buses stop every 20 minutes to take you to the subway station.Study the maps and it gets real easy to get around.The only drive we took was to Mt.Vernon and I recomend the small drive.Have fun.It's a daunting amount of walking.