A private school is a fee paying school. At secondary this is likely to be between 18-30 thousand per year per child. There might be selection tests for entry, but generally not as rigourous as Grammar School (although varies by school). A high achieving child might be offered a scolarship to reduce the fees. Some private schools have entry at 11+(yr7 grade 6) others at 13+ (yr 9 grade 8). Even so it is likely that the application deadline will be before April. Being private though you don't need to live here to apply, indeed many children attend private schools while their parents live abroad. If you think that your job might move again, and you can afford it, and you want a continuity of education it might be a good option. Rather confusingly some private schools are called public schools, but the general gist is that you need to pay.
A grammar school is a state school, so on the surface free. You have to pay for uniform, trips and might have to pay for travel to the school (this is the same for all state schools). You need to be in top 25% of ability based on English, Maths and reasoning scores in an exam for some, others (superselectives) take the highest scoring children. There isn't a set pass mark each year, it is standardised to select the top percent of children. Even if your ds passes then there would still need to be a space to get a place in the school. There are usually waiting lists. The usual time point (other than Cranbrook) for entry is at 11+ - so going into your grade 6 (our yr7). Beyond this it would be an in year admission application, but that is the same for all schools. For your dd she would sit the exam in Sept 16 (registration deadline end June 16). She would get a place if she passes exam and is near to school.
Other secondaries might be single sex or mixed and could have a faith requirement. This can vary from just being baptised to needing to be attending a church for 3 times a month for the preceeding 3 years. Again you can only get a place if there is a vacancy, and again there might be a waiting list. As the highest performing 25% have been creamed off already to Grammar Schools their exam results might not look as good, but a child can still do well if they apply themselves and they sit the same national exams at 16.
The council do have an obligation to find you a school, but it might not be the school of your choice or the nearest school. If the nearest school with a space is more than ?3 miles then they will help with transport costs ( to state schools, not private).