I had a history professor claim that many of the early Zionists terrorists eventually made their way into the Israeli government. It didn't sound right to me, is there any truth to the claim?
Short answer: yes, deep ones.
There were three main Zionist paramilitary groups active during the period of the British Mandate for Palestine (1920-1948):
The objectives of each of these groups varied over time — the Haganah's roots were as a self-defence force for the first waves of Jewish immigrants/kibbutzim, for example. The commonality is that at various times, particularly in the later Mandate period, all three engaged in guerrilla warfare against British rule with the ultimate objective of establishing a Jewish state in Palestine.
These groups are all frequently (though controversially — especially in the case of the Haganah) described as 'terrorist' groups. Their tactics variously included bombings, assassinations and other forms of guerrilla warfare that are familiar characteristics of insurgent movements. Some of the most famous examples of Zionist paramilitary action in this period include:
The relationships between these three groups, the Jewish Agency for Palestine, the Jewish National Council and the Zionist Organisation are incredibly complicated; political inclinations and objectives evolved and shifted over time. For some, a political/diplomatic pathway to statehood became the preferred option; for others, it seemed that the Jewish state would have to be created by force.
From 1945-1946, the three paramilitary groups worked together under the umbrella of the short-lived Jewish Resistance Movement. The relationship between these groups and the Mandatory Authorities is equally difficult — for example, the Haganah cooperated with the British authorities against the Irgun.
In any case, there's a large and complex historiography about the history of Jewish immigration, insurgency and state-building in Palestine.
To get back to the question at hand: it is fair to draw connections between these groups and the State of Israel, because they were all, to various extents, involved in that process. Their lineages can be traced deep into modern Israeli politics and society — particularly in the Israeli Defence Forces, which have a strong Haganah heritage. As I suggested earlier: describing the Haganah, the mainstream paramilitary force in the resistance movement, as a terrorist organisation is tendentious. But it's difficult not to reach that conclusion about the markedly more extreme Irgun and Lehi.
You can find an extremely long list of prime ministers, cabinet members, senior IDF commanders and Knesset members who were involved in the paramilitary resistance. Most were involved with the Haganah, but there have been two prime ministers who were involved with Irgun or Lehi: Menachim Begin and Yitzhak Shamir. In fact, the Irgun are the forebears of the Likud party (who are currently in government.)
Your professor was most likely referring to two of the more notorious politicians who made it to Prime Minister: Yitzhak Shamir and Menachem Begin. Both had ties to groups called Irgun and Lehi, who were nationalist-separatist right-wing groups willing to use violence to achieve political aims. It is widely agreed that they committed terrorist acts prior to and during the 1948 war, or that their men did under their orders. Some of the more infamous acts include Irgun bombing the King David Hotel (still the deadliest bombing to this day in the area to my knowledge). Menachem Begin claims Irgun had warned people inside to evacuate, but I don't think any historian really believes that all too much. Lehi is more notorious for assassinations, killing Lord Moyne in 1944 and Swedish Diplomat and UN mediator Folke Bernadotte in 1948.
Both Shamir and Begin were elected long, long after their time as leaders of these movements. In fact, right-wing politicians did not control a government until 1977, which is when Begin became prime minister. He promptly went on to make peace with Egypt. Shamir was a notable prime minister for his time during the Madrid Conference: he is widely regarded as something of a stubborn man, and he was extraordinarily intransigent and hostile in my opinion during negotiations and while talking to everyone, including the Americans.
That's a basic rundown of the most common accusations of terrorist leadership. Shamir left office in 1992 and was followed by Yitzhak Rabin, who signed the Oslo Accords I a year after being elected.