I would get the existing brakes set up nicely before starting to replace parts. You will probably find the braking is fine......
+100.
If you still have the stock Weinmann Calipers and levers, you will find they're quite adequate especially with your alloy rims and Kool Stops.
BUT as with any brake, they have to be set up correctly.
They only thing I would add to jyl's advice is to replace the housings as well as the cables - and finish the turkey-wing-ectomy while you are at it. (Take the (usually red) pivot out of the lever body and cut it flush at the shoulder, so they won't protrude into the hoods.)
You have a nice bike there - well worth some TLC!