A baby tooth brush with extra soft bristles is ok for stainless steel. Not for gold.
A horsehair natural brush is good for both stainless steel and gold.
Gentle brushing and NOT like you are trying to shine an army boot.

Dishing washing soap mixed with water or carpet foam shampoo is fine.

I always keep the crown and pushers side facing down so that water and soap run off flows always from the case.
I shield the water from the tap when rinsing off by holding the watch case in a closed fist and the water from the tap hits my fist and not the case directly.
I even use a misting bottle spray to rinse off the water.
I pay dry with a paper towel. Do not wipe with the paper towel.
For bracelet, if you can remove your bracelet, you can use an ultrasonic machine to clean. They can be had now for usd 10/-
If not convenient to remove, use the same method to clean as for the case.
You can also use a waterpik for teeth cleaning to target the bracelet to get into the recesses. Do not aim at the case. Hold the case with a clenched fist 🤜 and it is safe.
🤝