A simple example:
// item is passed in as a DOM object reference
item.innerHTML = "<div>html between this element was changed including adding a div</div>";
// this method adds a DOM child element to the element passed in
// usually the element passed into the function is with the keyword "this"
// ie: onClick="domMethod(this);"
// create div element
var div = document.createElement("div");
// makes the id attribute of the new element "newDiv"
div.id = 'newDiv';
// changes all the html between the new div element
div.innerHTML = "html between this element was changed";
// append element to item
// sometimes you want to pass into the function an ID of a DOM element instead of a reference
// using the element ID, a reference to the element is made
var item = document.getElementById(itemID);
item.innerHTML = "html between this element was changed";
There are others which are easy enough to find. You'll also find that creating a fully AJAX enabled website means you're going to come across issues with using the back, refresh, and bookmarking those pages. You're going to have to find alternative ways to use those old style web page functions.