You need a city with good public transportation if you don't want to rent a car.
This rules out much of the west.
What remains is Boston, Washington DC, Chicago, and Philadelphia. All of these are nice for a couple of weeks.
If I were you I'd go to Chicago. You can get around the city easily by train, and you can leave the city to explore interesting locations (for hiking, architecture, kayaking) away from the city by commuter train and longer distance rail. Local trains are the El (and there are buses too). Commuter trains are the Metra system, with several lines going to suburban destinations far from the city, and you can use the South Shore Line to travel to the Indiana Dunes or further, or take Amtrak to Milwaukee.
There is a lot to do in the city itself. Beaches are shut after the first week of September, but the entire lakefront is open to the public for walking, biking, skating, running, picnics... The city has lots of museums, art, music, attractions like the Shedd Aquarium, the Adler Planetarium, and the Field Museum. Architecture tours on the river, kayaking, gangster tours... Food is a highlight of Chicago. Lots of neighborhoods to explore too, and good shopping. You can also get to both airports by local train.
The other destination I'd suggest is DC. Entry to museums is free (though you need to book tickets well in advance for the Holocaust Museum and the Museum of African American history). Public transport is good. You can easily get to Baltimore / the ocean on the train or take a day trip to NYC by train.
September is a nice time to visit the US as kids are all back in school by the end of August (though Chicago public schools usually go back after Labour Day).