Most primary schools in Harare require parents to register their child for Grade 1 at the beginning of the year in which their child turns 3. There are some exceptions, however, such as Chisipite Junior School, where parents need to register their daughters even before they're born (or as early on as possible)! Usually a small administration fee is payable.
Bearing in mind the high demand for good schools in Harare, it is strongly recommended that you put your child's name down at more than one school, as there is a possibility they may not be accepted into their first choice school.
Nursery schools don't appear to be as stringent in their requirements, although it's a good idea to try and put your child's name down at least a couple of years ahead of the year in which you'd like them to start.
Some schools expect you to make an appointment to register, so it's advisable to ring the school before visiting.