Originally Posted by tuckervill
So if I don't want to retrofit the cranks, that means new chainrings attached to new cranks, which then means a new BB, right?
Not necessarily. I'm assuming it's a square taper spindle? There are lots of vintage and new cranksets that will fit square taper. There are also new sealed square taper BB available for cheap - no PITA adjustments necessary!
I know the old BB is British 68 mm. But what does the other measurement refer to? 108 mm, 113 mm, etc.? Which one is appropriate here and how do I determine that?
The other measurment is the spindle length, which will determine how far out the driveside crank sits. This affects clearance of the chainrings on the stay, and chainline, which is basically how close the chain is to running in a plane between the chainrings and rear cogs. You want as little skewing as you can get to maximize drivetrain life and improve performance. Generally, 108 is for doubles, and 113 is for triples, though those numbers vary.
Also, how do I determine which length of cranks is appropriate? The old ones are 190 mm from end to end but I don't know if they're measured from center to center.
I don't think it's 190. You're right, it's measured center of pedal to center of spindle. 170mm is the most common length. Some run 165 or 175 or some odd numbers in between. Most people don't see much of a difference within this range. Some do. It's your call.
Here are some Sheldon Brown webpages with related information.
This is a database of BB sizes. Good if you are buying a used crankset, as many are included here.
Some info on chainline
here.