Five and a half years ago, it wasbecame time for my daughter to go to primary school.

This sentence is a little unnatural to say. A few more common ways to say it might be:

"Five and a half years ago, my daughter started primary school." (This is probably the most common way to say it)
"Five and a half years ago, my daughter got/became old enough to go to primary school."

To give her a better care/attention, my wife quit her job.

Early this month, she found a new job, s. She was really happy and worked very hard, although she couldn't earn much with this job.

Step by stepBit by bit, she devoted less and less time to housework, and I had to devote more time to it.

"Step by step" is more for something like "solving a complicated puzzle," or "making progress towards a tough goal." e.g.

"Step by step, he got closer to completing the dungeon."
"Following this step-by-step guide helped me to complete the game!"
"Step by step, she got closer to graduating."

Five and a half years ago, it was time for my daughter to go to primary school.

"go to" -> "enter" also works

Another way to state this:
"Five and a half years ago, the time came for my daughter to enter/go to primary school."

To give her a better care, my wife quit her job.

Or: "To care for her better, my wife quit her job."

It might be better to give more context as to what kind of care you are referring to. Is it to drive your daughter to and from school? To help her with homework/studies?

Early this month, she found a new job, s. She was really happy and worked very hard, although she couldn't earn much with this job.

"though" sounds a bit more natural than "although" to me.

I really have no idea whether I should be happy or sad.