Updated: Oct 2, 2020
Looking at the entire cleaning regime, best is to break it down into daily, weekly and monthly activities. We have mainly three things to worry about:
Coffee oils - they make coffee taste stale
Mineral deposits, aka scales from hot water - they can clog up your coffeemaker
Fungal/bacterial growth in water storing areas - everyone knows what that can do
Any part which you can completely rinse out with water and wipe dry is fine to be washed with dish soap, otherwise best to avoid it completely - you can get by with water rinses and the other two cleaning compounds just fine.
If the water container in your coffee maker is detachable, wash with dish soap and dry out completely at least once a month for disinfection. I also use a disinfectant vegetable wash liquid while doing so.
For coffee oil deposits, eg in steel moka pot or drip coffee metal filters etc buy some espresso machine cleaning tablets like Cafiza, and soak in warm water overnight and rinse with fresh water once a month.
As for parts which collect mineral deposits from water, either buy coffee machine descaling powder/tablets and follow instructions, or just use a solution of 1:0 ratio of water and food grade citric acid crystals - approx one tablespoon in 1 litre of hot or warm water. Leave in the coffee maker for about 30 mins and then rinse with fresh water. This also works very well for electric kettle used for boiling water.
Other than these periodic maintenance rituals, for daily cleaning we just use water.