What is the rear spacing? If it's 130 mm hub width(inside the frame), you can throw a modern wheelsset in there. That will be your best upgrade. You can also work on up grading the rest of your components, from there(save the old parts). When you've upgraded everything you see fit, buy the frame of your dreams & build this one back up the way it was. You'll have two great bikes but, there is a warning attached. You may be disappointed with than new frame & want to put all the stuff back on the bianchi. Nothing rides like steel & modern geometry doesn't have the same feel.
Looks like you have a 6 speed frehub on the back. "Probably" 126mm width. You can still have a new set of rims laced to those hubs & new brakes. You may be stuck with downtube shifters, though(for the time being). It's still a worthy investment to upgrade what you can & save for that new frame, when you figure out what it is.