If you're an Israeli resident (i.e.: permanent or temporary resident or an Israeli citizen living in the country) then you're taxed by the Israeli Tax Authority on your worldwide income. Otherwise, you're only taxed by Israel on the income sourced to Israel (earned while in the country or from investments in Israel).
Regardless of that you're taxed on your worldwide income by the IRS.
There's a tax treaty between the US and Israel, which would probably not apply to you for most parts if you're a dual citizen, but some provisions may escape the savings clause.
Both countries have foreign tax credit, the US also has the earned income exclusion that may be useful. Keep in mind that some income that is considered tax-free in Israel, is taxed by the US. Specifically, your employer's retirement/savings contributions are going to be taxed by the US as current income.
There are plenty of US-licensed tax advisers in Israel, the US embassy once had a list, you can try to look on the website if it is still there.
A list of abbreviations to look up and get scared: PFIC, FBAR, FEIE, FTC, FATCA. Forms: 3520, 3520A, 8938, 5471, 114.