The bad news, that there is no new debug capabilities in comparison to WP7/IE9. Please take a look on How do I debug Internet Explorer on Windows Phone 7? since we are in exactly the same situation on WP8.
What I personally use on daily basis
- Debug your app in IE10 Desktop as much as possible 
- Weinre remote debugger. Demo video. You can use the following app based on Weinre to simplify its usage (no local setup needed) - IeMobileDebugger src or link to Store - Supports - Html traversing
Html node styles, properties, metrics
Reading console output
Executing js on device side from console (including intellisense)
Dynamic script injection - ability to debug live sites - Not supported - js breakpoints 
- For javascript line by line debugging use aardwolf. Demo with VS integration. 
- To redirect console trace to Visual Studio output and be able to use console.log("some message") for tracing 
index.html:
<script type="text/javascript">
    window.console = {
        log: function (str) { window.external.Notify(str); }
    };
    // output errors to console log
    window.onerror = function (e) {
        console.log("window.onerror ::" + JSON.stringify(e));
    };
    console.log("Installed console !");
</script>
MainPage.xaml.cs
private void Browser_Loaded(object sender, RoutedEventArgs e)
{
    Browser.IsScriptEnabled = true;
    // Add your URL here
    Browser.Navigate(new Uri(MainUri, UriKind.Relative));
    Browser.ScriptNotify += (s, arg) =>
    {
        Debug.WriteLine(arg.Value);
    };           
}