As an intermediate step, or until you can get something better, Craftsman has a HEPA filter cartridge ($30 to $35, I don't remember exactly) for their 5 gallon and larger wet/dry shop vacs. The HEPA filter should stop any fine dust from getting through a shop vac which could help with your immediate dust problem.
There are commercial dust hoods specifically for lathes, and several types of dust hood pickups with adjustable stands and articulating heads that attach to a dust collection system or shop vac and set behind the lathe to gather dust at the source.
If you are using a shop vac, there are cyclone lids for small barrels or 5 gallon buckets, like the Dust Deputy, which are supposed to trap the larger particles.
And wear a good dust mask yourself, not one of those paper things. Lots of the woods we turn are toxic in varying degrees, especially as fine dust from sanding.
Cheers,
Stephen B.
Tulsa, Oklahoma