Private schools are somewhat limited in Madinah. In fact it's one of the reasons why some Expats leave. Try searching
http://www.edarabia.com/schools/madina/. Minarat School follow a British curriculum.
Cost of living: a good quality 3 bedroom apartment with 2 sitting rooms would be SR30k - SR35k per year. Depends on the area of course. That's your major expense. Shopping and eating out depends on your life style but generally it's little bit cheaper than the UK. It all depends on what you buy and where you go. McDonalds, Pizza Hut price is similar to UK - whilst a local sandwich shop is cheap. Pepsi is 25p. Bottle of water (330ml) is 16p. 6 bread rolls - 16p. A bottle of 2 liter milk - £1.16. Go to
www.panda.com.sa to see grocery prices.
Petrol is only 7.5p or 10p per liter depending on what octane your car takes. Either way, you don't worry about filling up your car!
Medical must, by law, be provided by the employer.
Madinah is a much smaller city than Jeddah or Riyadh so accordingly the options for everything are less. That said, over the last 5 years 3 major malls have opened with good facilities (shops, restaurants, kids fun zone).
The atmosphere in Madinah is quite relaxed and it's easy to get about with a decent set of rings roads and continuous infrastructure development. The new railway station is under construction and the airport is only 15 mins away from the town center.