So if I had to narrow it down to the three most important stages, I'd put them right around the 1948 war (during the first ceasefire), the 1950s (France alliance), and post-1967 (United States alliance).

The transition, as you can imagine, was a painful one. I'll start off by talking about the first Arab-Israeli war, the 1948 War (or in Israel, the War of Independence). The First Truce, during the 1948 War (truce lasting from 11 June 1948 to 8 July 1948) provided the Israeli troops and formations extraordinary help. As Moshe Carmel (commander of the Carmeli Brigade) said, "The truce came down upon us like dew from heaven. The formations are tired, weary." It's important to note that despite there being a draft of sorts put in place by the Yishuv (Jewish leadership in Palestine), the forces were not professional and had to be consolidated to form the IDF over time, which began on May 26, 1948.

During this truce, the Israelis began the first step towards developing the professionalism we associate them with today. The Haganah, which had already begun to transition from an underground militia into a full-fledged army (and would form a backbone for the IDF), strengthened its control and weaponry apparatuses. Between May 5 and July 9, the IDF saw a doubling in manpower (30-35 thousand to 65 thousand), but the significance of this was that up to 4,000 were veterans of Allied armies from WWII. Specialists and veterans contributed greatly to the reorganizing process, helping take sorely needed command posts that would provide better leadership throughout the IDF.

This was compounded by, in what would become one of the most important strokes of the war, the influx of arms during the truce. By the end of the truce, over 25,000 rifles, 5,000 machine guns, and 50 million bullets reached the IDF from Czechoslovakia. These arms were extraordinarily useful in updating the IDF's military capability, and providing more opportunity to train and equip recruits. During these weeks, the IDF also started to get heavy weapons purchased from the United States and Western Europe, deploying some 30 Swiss-made 20mm cannons, a significant number of Krupp 75mm cannon, and even a few fighter aircraft from the Czechs again (25 by the end of July, and Spitfires began coming in September and October). The Israelis even managed to get ahold of transport aircraft and 3 B-17 bombers during this truce.

To comment more on how this all played into the modernization of the IDF at the time, the IDF beefed up brigades, and actually managed (with the new recruits) to fill out companies and battalions to actual standard size, more or less. Two new brigades were added to the roster, newly arrived armored vehicles began getting deployed, and ammunition was dispersed far more efficiently. As Morris puts it in his book on the war, "The army that confronted the Arab states on 8-9 July was radically different from, and far stronger than, that which they had met on 15 May".

The above process on reorganization began with the "Order of National Structure", issued by the Haganah Command in November 1947. The order established the Haganah as a national army, which would target other armed forces. The elite Palmach units were also integrated into this force, and they were organized into 9 brigades with 2-5 battalions each (the original plan was 4 brigades with 15 battalions). The formation of these brigades, and the appointment of commanders for them, was pretty difficult. Until that point, the largest formation had been a company, though in reality the basic unit was a platoon at that time. The lack of experienced commanders was clear here, and that played a huge part in how difficult it was to train new commanders: there was hardly any base to build from. However, there were also boons to this. Because of the lack of experience, the only experience Haganah forces were building on came from their experiences fighting Palestinians already. As a result, the Haganah encouraged improvisation, extensive commander responsibility, and a huge degree of independence among commanders. The Haganah also bought heavily into the "cult of the offense" as they called it (and you may recognize the term from other wars!), where they would head out to meet forces head-on rather than wait for the forces to reach their positions. The latter, this "cult of the offense" plan, was not necessarily helpful. It was merely the best Haganah doctrine could muster, because of the lack of experience. These doctrinal failures would have to be addressed, because fighting the Palestinians was decidedly not the same as fighting the Arab armies.

One of the deciding points of this reorganization effort of the Haganah was the restructuring of the manpower. Though it had 35,000 members, only 2,000 were full-time soldiers around November 1947. By mid-May 1948, the number of armed and full-time members of the forces had jumped to the 35,000 I mentioned earlier. When the Haganah became the IDF in May of 1948, we got the backbone of an underfunded military with undertrained officers that would morph over the years with more experience, training from veteran immigrants, and more modern weaponry bought.

Even back this far, the IDF forces had begun to get involved in the realm of education, which helped increase the morale of IDF troops and its involvement in everyday life. As quoted in a Knesset debate:

Although this instrument of state was not created without conflict as to its structure and functions, it was decided that the defense forces were to be small and professional, with a large reserve system...it should adopt the most recent scientific and technological innovations, inculcate the spirit of the Haganah and the Palmach in the new conscripts, act as a citizen-maker and aid in the integration and Israelization of the new immigrants, pay close attention to agriculture and participate in agricultural settlement efforts, serve as a model of fraternity, and associate itself with the youth movement.

The IDF, therefore, became such a huge portion of everyone's life from the start that the professionalization of the forces became part of every citizen's duty. The draft, in that sense, has been a citizenship and nation-builder, and (if you'll permit me to make a small anecdotal reference), it's often referred to as "the place you make friends for life and become an Israeli" by many Israelis I know. Further, the "small" nature provided personal relationships that developed very closely within the IDF, providing more motivation. The IDF is very centralized and concentrated as a force, and this would play a crucial role in further conflicts as well when communication and central organization would play a large role in Israeli success.

This, as far as I'd put it, marks the first stage of modernization. Now we skip forward, past the end of Israel's success in that war, to the French alliance. The Israelis, after the war, were facing a fairly large deficit in terms of continuing modernization of the IDF compared to their Egyptian neighbors. In the early 1950s, there were still significant quantitative and qualitative advantages for the Arabs in arms, and a concerted effort by the Arabs would have been very dangerous for Israel. As Ben-Gurion put it, "I could not sleep at night, not even one second. I had one fear in my heart: a combined attack by all the Arab armies". In light of this, Ben-Gurion undertook yet another attempt to modernize the IDF, but this time he wanted to work with a great power to do so.

Naturally, he turned to the United States first, hoping their quick recognition of the Jewish State (11 minutes for de facto recognition!) would provide the basis for an alliance. However, the United States turned him down. They continued to adhere to the arms embargo imposed on the region in 1947 (during the Civil War period preceding the War of Independence) even though pre-existing agreements meant Arab states were still getting weapons and training from the powers involved.

Continued below!

As follow up question,does having a conscript army makes it more difficult in making an army well trained and professional?(in Israel ofc)