Children born in Israel are automatically considered Israeli citizens. But there are forms to fill and papers to sign.
The hospital will issue several documents when your baby is born that you can take to misrad hapnim (Ministry of the Interior). There are misrad hapnim offices in Afula, Akko, Ariel, Ashdod, Ashkelon, Bat Yam, Be'er Sheva, Beit Shemesh, Bnei Brak, Eilat, Hadera, Haifa, Herzliya, Holon, Jerusalem (Grinbaum Square, near the Knesset), Jerusalem (Gilo), Karmiel, Kfar Saba, Kiriyat Shmona, Ma'ale Adumim, Modi'in, Naharia, Natzeret Illit, Netanya, Petach Tikva, Ramat Gan, Ramle, Rechovot, Rishon LeZion, Tel Aviv (across from Azrieli), Tiveria, Tzfat. Office hours vary and you can only go to the misrad hapnim of your residency. When in doubt, go to the office that issued your teudat zehut (ID card).
Bring the teudat leida chai (temporary birth certificate) and teudat shichror (physician release) that the hospital gave you as well as your teudat zehut (ID card) and your husband's teudat zehut (if you are married) and any passports you and your husband may have to misrad hapnim. Your baby need only be present if you want her/his passport issued at the same time. You can always get a passport later. Just remember that there is no legal way to take a child out of the country without a passport.
Misrad hapnim will issue the teudat leida (birth certificate) and update the dependent information on your teudat zehut. Birth certificates are in Hebrew but you can get one in English as well. If your child has a Hebrew and English name, check the English spelling carefully as common European names are often spelt differently in Israel. It will take a few weeks before the teudat leida is issued but the teudat leida chai serves the same purpose in the meantime. Passports take about a week.
If the hospital did not automatically inform bituach le'umi (national insurance) then you can take the hospital information (teudat leida chai, teudat shichror) to your local office or wait until the teudat leida is issued by misrad hapnim. All registered children are automatically insured until they reach 18. Sooner is better as bituach le'umi issues your ma'anak leida (birth stipend), kitzvat yeladim (monthly stipend) and deals with maternity leave; generally 6 to 14 weeks of paid leave. They used to issue cheques but now everything is deposited directly into the mother's bank account.
If you are in the country on a temporary visa then misrad hapnim will add your child to your visa. Kitzvat yeladim will expire when your visa expires but it can be renewed if you renew your visa and they will pay any benefits you may have missed in between visas.
The United States, Canada and many European countries require an Israeli teudat leida before you can register your child as a citizen of your home country. Other countries may have other rules but it is illegal to register a child born in Israel as a foreign citizen without any official Israeli documentation.
Bring your child's teudat leida and your passport to your home country's embassy or consulate and they will issue a passport according to their rules. The British embassy will issue a British style birth certificate. Most embassies require an appointment and the American embassy will not even give you the time of day without one.
You can take your child out of Israel once she or he has an Israeli passport but it will be far easier to bring the child into your home country after you have that passport.