I have purchased a truvativ bottom bracket adapter form my early 70's schwinn continental. I have also decided that I would like to purchase a used vintage shimano 600 crankset. They seem to be easy to come by online. I have been doing some research online about bottom bracket sizing, and I want to make sure I purchase the right parts. Does anyone know what size bottom bracket axle I would need to keep my chain in line after the conversion? I know I need to purchase a 600 series BB if I am using that series crankset. I would like to go with sealed bearings. I'm just concerned about keeping thing symmetrical on my vintage bike. Please any advice would be great.

Which model 600 cranks? different years/models used different lenght spindles.

It doesn't have to be a 600 specifically any shimano or sugino the correct lenght and offset will do. A 600 will just be better quality thana rx100 or 15

You don't have to use a 600 series BB, nor even a Shimano one.
I'm assuming the adapter converts to English (1.37x24) threading, and you'll need to select the BB by the spindle length required for your crank.
A good value/priced cartridge unit is the Shimano UN-5x series, generally priced from $20-$30.

6400 is what I am looking for. I just did a little more research that told me the spindle length is 113 mm and the chain line is 43.5 mm. I guess my question is will the keep things in line on such an old bike or will I need to add a ring to my back wheel? So many questions...

So as long as my dimensions match the crank I can go with what ever BB I would like? Ok this is making more sense.

6400 crank should use the 113mm spindle length with your 68mm BB shell (Italian/70mm uses a 115mm).
The 43.5mm chainline (center of the two rings) is the same irrespective of the number of cogs in the back.

Thanks! This has really clarified things for me. I have things all figured out now!