Heya,
Not to be rude but the very best tip you'll ever receive - EVER is search is your friend. With that tip all you need to know is what to search for and you'll most likely find it. For instance if you googled php check email script or something similar you might find what you're looking for.
To check email addresses I'd use a regular expression and for the activation it's pretty straight forward...
Basically you have a DB table field that is named "activated" or something similar. By default the table is entered with the value "0" or "no" etc. After the registration use a script to send an email to the email address. The email should have a link with dynamic URL parameter of registered users id/foreign key/etc. that goes to a page (activate.php) with an UPDATE query that updates the record based on the URL parameter. If you take this one step at a time it's actually very simple to setup. The activate.php will have a filtered query that updates table field "activated" where variable for $_GET['url_param'] = DB for registered users id/foreign key/etc. This will find the record for the user that is being activated through their id and update their activated table field from 0" or "no" to "1" or "yes". Sorry I couldn't describe more but I am too busy to write it all out and it should be enough to go on anyway. The URL parameter could be circumvented by having someone simply visit the URL to activate a user so a randomly-generated foreign key is recommended for the URL parameter on the activate.php page.