How can I get Autocomplete behave as in most of the sites out there?
I think this is a JQuery problem rather than a Drupal problem.

You should be able to add a JavaScript function, something like the following:

$("#searchField").select(function(event, ui) { 
    $("#searchField").val(ui.item.label);
    $("#searchForm").submit(); 
});


It's necessary to assign the value before submitting the form because if the user clicked on a list entry with the mouse, the value won't have been assigned before the select event occurs.

I haven't actually tested this, but hopefully this will send you in the right direction.