Well, since you're asking ...

I wouldn't want to change the look & feel of the bike, so no major surgery and certainly no Campagnolo Nuovo Record or other fancy stuff. Here's what I would do:

1. Remove some superfluous steel
The spoke protector and chain guard need to go. This will improve the looks and the weight at no cost and little work (if you do it yourself; little cost and no work if you have your LBS do it.)

2. Move the shifters from the stem to the down tube
You'll need clamp-on shifter bosses, but the rest will work as-is. This will improve the looks big-time and help shed (a little) more weight.

3. Remove the safety brake levers
You can either remove the 'turkey wings' (this will probably leave a stub) or get a pair of replacement levers (Weinmann or Dia Compe). They should be easy to find for little money. The weight saving is negligible, but it will improve the look. Get some rubber hoods while you're at it.

At this point he bike has been 'de-dorked' and will look much more grown-up. It's still a little heavy though. My next steps would be:

4. Replace the wheels and tires
Those look like steel rims. They look awesome, but weigh a ton and and the brakes won't work well - if at all - in the wet. If so, I'd be looking in my attic for a wheel set with aluminum rims. (and preferably high flange hubs for looks.) This will really save some weight and make a big difference in how the bike feels and rides. Get some good lightweight tires. Panaracer Pasela is the default choice, and they weigh half as much as a cheap tire. If the rims are 'hooked', get the folding version, as they're even lighter. The 28mm TourGuard is a nice compromise between weight, looks, comfort and durability.

5. Replace the crank set
If it were a traditional Stronglight or Nervar set I would leave it on, but in this case I'd replace them with a cheap aluminum set. I have a low-end but very light Thun Coronado set that would look just right on this bike. Sugino Maxy would also be a good choice, IMO.

The weight is now probably down to between 12 and 12.5 kilograms, and light enough for anything but the highest Alps.

The rest has more to do with personal preferences:

6. Replace the saddle
I'd want a black, slightly worn Brooks on it. Preferably a B17, so it will take a saddle bag when touring, which is what I'd be doing with it. I'd probably replace the seat post as well with an aluminum one, while I'm at it. Or maybe not, as I like the look of the traditional saddle clamp. Or find a Birmalux in the right size, that will let me have the best of both worlds ...

7. Replace the pedals and add toe clips
I don't know about you or your family members, but I like them.

8. Lose some sleep over the colors of the cable housing, toe straps and bar tape
This is generally the most difficult part. At this point I think I'd go for all black, as it sets off the shiny bits so beautifully. Cotton bar tape.