If you're saving them in a database, then yes, addslashes() will escape any dangerous characters for you and when you come to display it back in the browser just use stripslashes() so that users don't see ugly escape characters.
If you have magic quotes enabled in PHP then the server will automatically addslashes to all $_POST, $_GET and $_COOKIE data for you though, so it's worth checking if this is enabled first because escaping data twice will just give you headaches.
Paranoia is definitely a good trait where this stuff is concerned