You might have misunderstood.
I meant MT tapers and/or collets that fit in the spindle, not the TS. Need to know what the taper is in the bore of the spindle. So you can begin to watch for a collet set. Plus you need a spindle center for doing a lot of work that is best done between centers, rather than in a chuck. Using centers allows you to take work like shafts and threaded parts out of the lathe. So long as you do not remove the drivr/dog attached to the work, and put the tail back in the same place, you can try the work in say a larger assembly, and then put it back in the lathe "perfectly" if it needs a bit more lathe work.
A 4 jaw chuck and a set of centers with dogs and a face plate to drive the dogs will be your most versatile work holding methods.
A true running collet insert and good collets would be next, for smaller diameter work. Way down the list, a decent 3 jaw chuck would be fast for things that can be done in one chucking, or that aren't particularly precise or critical.
smt