Israelites are the religious and ethnic group presumed to have descended from the 12 tribes of ancient Israel, which seem to have begun as federation of diverse nomadic tribes in the late Bronze Age. Modern Israelites include Jews and Samaritans.

Jews are variously defined as descendants of the Biblical character of Judah, of the ancient nation of Judah, or followers of the Jewish Torah... Often, the term is conflated with broader ancient groups like Israelites and Hebrews.

Israelis are the most specific group - simply citizens of the modern state of Israel.

First, it's important to note that you're referring to terminology in English which, historically, was not used in English. So usage can be a bit muddled.

Anyway, the oldest term is what we'd translate as "Israelite", which in Hebrew is simply the equivalent of "Israel". That's used today to refer to the group described in the bible, and the bible itself uses that term (along with בני ישראל, "children of Israel").

After a series of wars, the tribe of Judah, or in Hebrew יהודה yehudah became the dominant Israelite tribe. Because this was the name of the people as well as the tribe as well as the geographic area they inhabited, it became common to refer to all Israelites by this term (the book of Esther in the bible provides attestations of this). The name of the Persian province was Yehud Medinata, as well. It's this term that passed into English as "Jew", but the original terminology that would've been used at the time makes no distinction between "Judahite", "Judean" and "Jew".

However, unlike in English, the equivalent of "Jew" never quite replaced the equivalent of "Israelite". Jewish texts do still use "Israel" as the term. It also has an additional meaning, a person who is Jewish but not a member of either of the priestly groups (so the threefold division is between a Kohen, Levi, and Israel). When used in Jewish texts the equivalent of "Israelite" does not necessarily imply antiquity, that it references a time before Judean dominance changed the term--it can still be current. When the Talmud speaks of a שור של ישראל shur shel yisrael "ox of Israel" it is referring neither to the modern state of Israel (since that wouldn't exist for another couple thousand years) nor to the era of the bible specifically, but to what we'd call "the Jewish people". That goes for the term "Israeli" too, -i is a way of making adjectives.

But in English, "Israeli" is specifically used for citizens of the modern state of Israel, after 1948. It's not synonymous with "Israelite" in English, even though there's no distinction between these terms in Hebrew. "Jew" refers in English to the same people after Judah was the primary surviving tribe, but this can be roughly synonymous with "Israel" in Hebrew.

To confuse things further, there's another, more obscure term in English, "Israelian", which is of or pertaining to the biblical kingdom of Israel. One may speak of Israelian Hebrew, for example. Again, this is a distinction that exists only in English, not in the languages where these terms originated.