Training bras are for preteen girls to use while their breasts are still too small to fit into regular sizes, but are either: sliiiightly big enough to show through a shirt, or the girl is super-eager and wants to wear a bra as soon as possible to feel grown up.

Nothing more. I suppose it is 'training' to wear a 'real' bra.
I fell into the latter group, btw..
But, yeah. Training bras do not change the shape of the breasts. My breasts changed shape over the years naturally, as they stopped growing and the.. er.. 'contents' settled a little. The shape you have while they are growing/right after they've stopped growing is not the shape you will have after a year or two. Gravity does the work; bras do not. Unless, as has been mentioned, they are faaaar too restrictive.
As for taping.. I.. I don't see how that would help, to be honest..