Thanks a lot for the replies! So it seems wheels would be a good start (leaving me with the opportunity to swap them to the new bike too)... I've never even thought about how much of a difference an external bearing BB/crankset would make so I guess that could be step two.
Would I be better off just sourcing an ultegra crankset/BB or should I try to get an entire groupset? I don't know how much of a difference I'd see going from Shimano 600 (Ultegra, 6400) to modern Ultegra (say 6600)?