How to make that when the user clicks a submit button, some kind of result message appears and after few seconds redirect to another page ?
Sounds like javascript.
make a function that throws up an alert or shows a layer... do something with like say window.location after x num of secs has passed.
Alternately you can use a meta-tag to refresh to another page;
<meta http-equiv="refresh" content="12; url=http://www.site.com/page.php">
(12 being the amount of seconds before refresh)
(i think, can never remember the syntax😉
or, of course, header()
header("Location: http://foo.bar");