Is there a town on the train line between Edinburgh and Glasgow that would be convenient to get into both cities?
Falkirk is perfectly situated, although it's a bit closer to Edinburgh. There are several historic sites there, including part of the Antonine Wall, and the Kelpies and Falkirk Wheel are easily reached by public transportation if you don't have a car.
There are many restaurants and shops in the central area, as well as in the outlying areas. There are three train stations, although only Falkirk High serves the Glasgow to Edinburgh line. The closest hotels to the station are about a half mile away, or about a 15 minute walk downhill, 20 minutes back up.
You can also walk for miles along the Forth and Clyde Canal.
Hope that helps!
p.s.: Little known fact - the train service between Glasgow and Edinburgh is classless, so you can sit in the first class section and feel like a toff.
3 of us are planning to be in Scotland in September and are thinking of basing ourselves between Glasgow and Edinburgh (maybe in Falkirk, Bathgate or Armadale)--ideally in a B&B so we can be flexible enough to drive/take a bus or train to the cities or around the central part of the country. Any suggestions about how to access a list of B&B's?
Armadale and Bathgate are not particularly attractive towns. Falkirk, with its historic association, would be much better. You'd also be close to the Falkirk Wheel, the Kelpies, Callendar House, the Antonine Wall, and Falkirk High train station.
You can check out B&Bs on booking.com, and when you find one that piques your interest, check out the reviews on tripadvisor. If all else fails, the Premier Inn at Camelon (on the outskirts of Falkirk) is nicely situated on the bank on the Forth and Clyde Canal. It's about a 15 to 20 minute walk to the Wheel, and you wouldn't have to pay the three pounds to park at the Wheel.
Hope that helps!