I'm running halogens on a 500 (1969) with no worries as the original bulbs were like birthday cake candles. Just get bulbs with a wattage rating as close to the 126 bulbs as possible so you don't fry the wiring. I think later 126s had relays, which avoid the extra power running through the ignition and light switches - I strongly recommend fitting relays for that reason, If you want to fit relays its all very simple and I can produce a wiring diagram if you need one.
The alternator produces more than enough power.
If you go somewhere like Halfords, they'll have a guide by the bulbs which will show you which halogen bulbs to fit in a 126 - its the wattage that's important - the size is near universal for cheap old cars.
I am looking at brightening up the rear end as the lights crap there too. Anyone tried LEDs? They take less power than bulbs but are brighter. Never tried them though.
EDIT: heres a relay diagram from another site - let me know if you don't understand it. I'd put a fuse per relay and third relay so there are one for each dip or one blown fuse is lights out. Its easy enough to find relays with inbuilt fuses now.