There are a few factors at play here.

First, the Israeli army didn't go straight from nothing to superpower. Jewish paramilitaries had been arming and fighting Arab groups and the British since the 1920s, to varying extents. Their weapons and trained personel became the nucleus of the IDF. In fact, the IDF was basically formed by absorbing smaller paramilitaries into the largest and most reputable one, the Haganah.

But on the other side, Israel didn't go directly to superpower from nothing to a militarily powerful country. In 1948, Israel faced several countries (mostly Palestinian militias, Jordan, Syria, and Egypt), but these enemies were not united. In fact, Jordan actually forcibly disarmed Palestinian militias. By the time other countries joined the IDF had been fighting for years. Besides experience, this meant having logistics in place for getting weapons. Additionally, Israel was fighting for its existence, which improved morale. Israel had no choice but to win, but priorities were more complicated for its enemies.

By the end of the war, Israel was hardly a superpower major power. Israel won the Sinai War...sort of...but with the assistance of the UK and France. French assistance was particularly important for Israel in the 50s--the Air Force got Mirages, and most importantly, France assisted Israel in developing nuclear weapons. By the mid-60s Israel was better-equipped and better trained than the Arab countries it was at war with. Israel decisively won the war in 1967 for this reason, but the war was so overwhelmingly in Israel's favor mostly because of Israel's devastating first strike against Egypt, which gave the IAF complete air superiority in the whole region, and allowed Israel to fight Egypt, Jordan, and Syria more-or-less one at a time. After 1967 Israel began receiving substantial military support from the US, which helped resupply during the 1973 war. From their Israel had a huge material advantage over the Arab countries, which recieved much less substantial support from the USSR. Of course at that point it hardly would make difference anymore, since Israel did not fight another conventional war against the neighbor countries.

These threads have very in depth top answers.

How did Israel develop such an effective military so quickly after its independence?

When did the Israeli army develop its reputation for professionalism, and what was that process like?