At our school, all the little ones have a book they fill in, for the parents to date and to comment in when they read at home with their children. At the lower end 4-5 year olds, if they read with their parents at home, they get a stamp for each read and if they read 5 times in a week with a relative, they get a bag of flumps on Fridays and a sticker in assembly to encourage them to read more out of school.
As for the books, they are given a new reading book by the teacher as needed and then once a week they get to go to the school library and pick one book of their own choosing.
Jayne