I usually just cover leather parts with plastic bags or shower caps.
Then I give the bike a bounce, and the wheels a spin to get excess water off the bike.
I sometimes lube pivot points and exposed shifter/brake cables.
I use Chain-L No.5 chain lube, so I don't bother with relubing the chain after rain.
I am trying to see how it does with no relubing.

People say:
Clear silicone caulk the headset where stem enters (if threaded headset). Caulk top of the seat tube (where seatpost enters).