I live in a tropical country, so most of the year the temperature is above 20° Celsius, so coldness is not an issue here. It can also be unpredictable for rain.
I have tried both strategies: a) (try to) keep dry while riding under the rain and b) (act like) not caring about it.
With strategy a, the problem is sweat accumulates inside rainproof gear, so I end up as if had worn no rain gear at all.
With strategy b, it's all about what waits for me at the end of the ride. When I start and end my rides from my car or home, hotel room, etc, I just leave a second clothes change there, including socks and footwear. If that's not the case, I carry a light jacket and just wear it even on top of my soaked jersey after the ride. That is because for me the severe cooling of the jersey causes me a severe cold (as in sneezing, coughing, etc...). That works for me because I would remain on wet clothes for a couple of hours before having access to a warm shower and dry clothes.
If I had to commute under rainy conditions, I'd just roll a change of office clothing and an extra pair of shoes. When commuting I always aim to arrive with time to spare so I can go to a toilet stall to freshen up and change clothes, no matter if wet from rain or from sweat.
Regarding shoes specifically, I use MTB biking shoes that usually have breathing fabric on the sides. They get soaked to the point water spews out on every step or pedal stroke. I just don't care. Fungus does not develop "under water" so as long as I do not remain on wet shoes/socks for too long, there is no problem at all.
When circumstances allowed and I had to use the shoes next day, a hair drier works wonders. Also useful for socks and other small items.
Other times, for short rides (4 hours or less) I just did wear them as is and washed and dried them afterwards. It is awkward when you put them on, but as soon as the water in them reaches the same temperature as your feet, you barely remember they are wet.
I prefer to wear tight clothes that move with your skin, rather than loose clothing that rubs against it. Heavy rain has caught me riding with baggy "surfer" shorts and the wet fabric dragging on my thighs where a huge waste of energy, specially at the end of a tiring stage on a multi day ride. Also the wet clothes caused severe chauffing on the inner leg, near the groin. However, when using proper cycling shorts, even under heavy rain, I had no problem at all.
The same applies to shoes, I prefer to have them as tight as necessary so they do not rub against my skin. Constantly rubbing the wet skin with force and wet fabric is a recipe for blisters. As i said, my shoe's material is very breathable.