Nice <basic> Peugeot bike. These frames make good fixed wheel conversions
Your rims are old school but also indicative of an inexpensive wheel set. Steel rims are heavy, slower to get up to speed and rubbish in the wet -as you have found out!
These old Peugeots can be fiddly to upgrade because of non standard components fitted (Seatpost diameter , Headset threads). In addition your rear wheel spacing will need to be changed to accept more modern wheels. Also does your frame have a braze-on gear hanger or a stamped derailleur mount bracket? Perhaps even a Maillaird Helicomatic freewheel?
It is probably easier (and cheaper) to replace the consumables such as brake blocks, tyres, cables etc and just ride it and accept the bike as it is. I had an old Raleigh Arena GTX between the age of 15 -22 which had steel rims. I remember the original brake blocks had leather inserts to <improve> wet weather braking, I can't remember if they made any difference but I did 1000's of miles on that bike including touring with loaded panniers and you sort of get used to the poor wet weather braking!