I'm sure you've heard the analogy that children's brains are like sponges - they absorb things easily and quickly. This is so true for memorising the Quran.
When 3 and 4 year olds come to my toddler and prekindy classes, parents would worry that their children don't know how to recite the short surahs. The problem is that they have not expose their children to the recitation of the Quran.
To have children memorise the Quran easily, parents should recite the surah to their child every day.
When my children were babies, I would play Juz Amma for them while they slept. At night time, I would recite Ayatul Kursi and the three Quls. I would continue to do this every night then, when they were able to talk, they would join in the recitation. Gradually, they were able to do it themselves.
This is how children learn to memorise the Quran, by repeating over and over again. This was how I taught my children the other short surahs as well. In the morning, they would learn the next short story by listening to me reciting it. During the day, I would give them the Quran player and I would tell them that they had to listen to the surah being recited on it at least three times.
These days, my children are older and when they learn a new surah, they would use the ipad and iphone. I would tell them, for example, that this holiday our aim is to learn this surah or ayat. They would take the ipad and iphone and start listening to the first ayat, press repeat and listen over and over then try to memorise it. Once they have that memorised, they would go onto the next ayat and so on.
Subhanallah, Allah has made memorising the Quran easy. But you need to expose children to hearing it repeatedly every day.
Learning to memorise the dua is similar. You can read about at this site where I have written about it many years ago. It's called 'Saying It Out Loud - The Easy Way to Teach Children the Daily Duas'.
Start your children on the road to memorising the Quran today. It's never too early. Turn on the Quran and let them listen and repeat over and over.