While pushing each of my three sons on a swing when they were wee lads, I would count-by: 1,2,3,4,5,6,7,8,9,10; then 2,4,6,8,10,12,14,16,18,20; then 3,6,9 etc. The swing push would be over when we got to 100. My wife hung count-by sheets in the kitchen. We spoke about counting by eggs as if they were a fraction of a dozen: 1/12, 1/6, 1/4, 1/3, etc. I taught the youngest how to compute squares in his head when he was in 2nd grade. First, he learned one squared through 10 squared, then 10,20,30, etc squared. Then we played a game: what is 20 squared? what is two times 20? what is twenty squared plus two times twenty? what is twenty squared plus two times 20 plus 1? what is twenty-one squared? These exercises were in the car on a ten minute ride to school. We started to work through computing products as differences of squares.
Certainly I taught the boys some modular arithmetic, and they all attended the math circle --- even started them a bit too young.
Also, they were taught how to count to 1023 on 10 fingers. Lots of cute tricks. In terms of the mental calculations, even if you can't do the arithmetic quickly, you can teach the child to do so. When the child sees that you struggle with it, then (s)he has someone with whom (s)he can compete.
In addition, I would stress units and developing answers as complete sentences and guiding writing.
Read "Alice in Wonderland" and "A Wrinkle in Time" to the child at about 1st or 2nd grade. Emphasize the connections between math and human development.