To achieve a smooth navigation on your multipaged mobile html site using onsen UI, you have to do the following:
1) Insert the tag in the index.html page. This should be placed inside the tag example is shown below
<!DOCTYPE HTML>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1, user-scalable=no">
<meta http-equiv="Content-Security-Policy"
content="default-src * data:; style-src * 'unsafe-inline'; script-src * 'unsafe-inline' 'unsafe-eval'">
<script src="lib/onsenui/js/onsenui.min.js"></script>
<link rel="stylesheet" href="lib/onsenui/css/onsenui.css">
<link rel="stylesheet" href="lib/onsenui/css/onsen-css-components.css">
</head>
<body>
<ons-navigator id="navigator" var="navigator"></ons-navigator>
......
</body>
</html>
2) Your inner pages should not have a opening tag, or other similar tags, but should look somewhat as shown below (assuming this page is my-second-page.html)
<ons-page id="my-second-page">
.......
</ons-page>
3) To navigate from my-second-page to my-third-page.html, use the Onsen UI pushPage function as shown below. This is a javascript snippet that will be placed in your javasript file. This code will be executed when a user clicks a button to navigate to my-third-page.html
document.querySelector('#navigator').pushPage('my-third-page.html');