First of all you should plan the trip well in advance to Israel and have all the vaccinations done to you and your children, which are necessary in case of all countries you visit, not just Israel. However, you should be extra careful if your children are accompanying you and prepare them properly to face some unexpected climate changes.
You should read about the country, especially about any particular health issue that you can face there as a visitor at a particular time of the year. Also, try to know about the health services of the country, health insurance requirement and other necessary things. For example, you might be required to get a shot of any particular vaccination before heading to the country or medical certificate that the country might require you to produce at the time of entry. This requirement can be different from country to country.
While in the country adopt all health, safety and security measures to avoid any problem. For example, you should protect yourself from sun heat, dust and sand etc. Also, try to avoid eating unhealthy food items and drink a lot of water. Always buy safe drinking water, and do not take anything fizzy. Ask tour guides for any particular health risk that you can possibly face. Sometimes immune system gets weak and bacteria from a particular country can launch their attacks effectively on you. You can know about all these problems well before heading to the country and protect yourself.
Despite taking all precautionary measures if you still get ill, try to check with a local doctor immediately. Do not delay the matter as a serious illness can cost you more. If you have got health insurance, you can be paid by the insurer, if not you will have to afford the bill yourself. In any case do not avoid visiting a doctor. Since you are most likely to be alone in the country, away from your home and family, the effect of the problem can be multiplied quickly and you might find very hard to fight off the problem. If you think you cannot cope with the problem alone, better head back to your home country.