I don’t have three children clamouring to be fed, but one thing I have found really helps me to cook quickly is to use frozen food ingredients, so in my freezer I always have
chopped white onions,
chopped red onions
frozen garlic cubes
frozen ginger cubes
frozen peppers
frozen mash
frozen roasties
frozen yorkshires
other frozen vegetables are also available
i also buy blocks of cheese on special offer, then grate and freeze.
if you are not a very efficient cook ( I am a good cook but I am also lazy!) then it is often the prep that takes the time, buying frozen preprepped veg is obviously more expensive than fresh ( except for frozen garlic which is ridiculously cheap) but it is cheaper than takeaways and saves you precious time.
The other thing I would say is take the time to learn to make a good basic white sauce preferably with cheese in it, and a basic tomato sauce. You will then have the skills to make a huge number of dishes, cheese sauce for macaroni cheese, veg lasagne, or cauliflower cheese, tomato sauce to make spag bol, or to cook or serve with meatballs, fish or chicken.
Also , learn to cook rice, I use Jamie Oliver’s method which gives cooked rice in ten minutes with no faffing, just remember to set the timer and not to peep.