How Old Does Kitten Have To Be Spayed?

I have a cat that is 7 years old. She has had no accidents for the last 4 years and she does not go outside so I do not think it’s time to spay her now. However, I am thinking about getting another kitten in 8-9 months because my mom said that cats are more active then kittens which means my new kitten would be spayed when he/she was around 1 year or younger? Is this true? Also, what age should I wait till to get them fixed if they will start going through puberty at 3 months of age? Please help me!