When the state was established, the date of the Declaration of Independence in 1948, the 5th of Iyar, was set as Independence Day, an obvious and natural decision, parallel to the practice in most modern nations. And the actual observance of the day also paralleled that in other states: picnics, parades - with an emphasis on the military - and fireworks displays. In "the good old days," everyone converged on the downtowns of the cities at night on the eve of Independence Day, and there was folkdancing all night, as well as general milling around and eating. There is still plenty of milling around and eating (though not this year, as many people were afraid to congregate in public places), but folkdancing is no longer the central ritual of the the civil religion of Israel that it once was.
Similarly, the picnic, while still a ritual, has lost some of its original meaning for many people: I think originally the excursion to a historical or nature site carried a sense of getting to know the land, "this land is our land," a celebration of the beauty and the rich history of the new state. This has not been lost completely, but for many people, the barbecue has become central, the travel less so. Shopkeepers who specialize in seasonal wares sell plastic scouring pads before Pesach, paper decorations before Sukkot, and charcoal before Independence Day.
Military parades, once a mainstay of the day, have become passe, probably expensive, maybe uncomfortable in terms of our image of ourselves. We no longer do them. But army bases have special open houses on Independence Day, and there are tours and demonstrations; little boys can play in the tanks that in a few years they will be driving. Parents can reminisce for their kids...
An interesting aspect of the ritual that may be unique to Israel was the decision to set Memorial Day as the day preceding Independence Day. On the one hand, it makes a lot of sense: after all, it is important to remember that we did not get this state "on a silver platter," but that it came - and comes - at a very high price. On the other hand, emotionally, the juxtaposition and transition are difficult. On Memorial Day eve and day there are sad ceremonies of commemoration, and they are not of course limited to those whose loved ones died in the dimly remembered past. If even I, a new immigrant without extensive networks of family, school, and army, encounter among my friends and neighbors the widows and orphans and bereaved parents generated by the ongoing struggle for existence here, then clearly no one is outside the circle. And everyone is in the army, or has a kid in the army, or has a kid who will be in the army. So the sad songs in the ceremonies are really sad, for everyone, and not just patriotic rituals. Few question the policy requiring that places of entertainment be closed on Memorial Day, and forbidding celebrations (though it's kind of amazing - can you imagine such a prohibition in America?). When the day ends, at sunset, the mood changes abruptly from mourning to celebration. It's an odd feeling, and I can't decide whether I think it was a good idea or not.
The religious observance of Yom Ha'atzma'ut is the litmus test for distinguishing the Zionist from the non-Zionist orthodox: for Zionist orthodoxy, Independence Day is a religious holiday, a celebration of God's beginning our redemption by the miraculous establishment and preservation of Israel. Hence, there is a special service that morning, with Torah reading (a popular day for bar mitzvahs) and the recitation of Hallel psalms (traditionally recited only on biblical holidays and Chanukah).
This radical decision to equate Israel Independence Day with the divinely ordained festivals is not accepted by the non-Zionist orthodox (the "haredim," or "ultra-orthodox"), who assign no religious significance to this day.
Interesting to consider: what is this day to Diaspora Reform Jews? just another historical milestone among many, as the ultra-orthodox insist? or a unique, messianic event, evidence of God's hand in history, as the Zionist orthodox believe? Or are there other options?