Not many steps here. If it were me I would:
Remove the any allow all rule on the interface for the subnet in question.
Add a rule to allow DNS to the interface IP.
Create an alias containing the IP addresses of the sites you want to allow.
Add a rule to pass traffic from the subnet to that alias for TCP.
If you really wanted to restrict further use a ports alias to allow only ports 80 and 443 as the destination too.
BUT... that will only work well for sites that resolve to a single IP address or only if you have all the resolvable IPs in the alias. So it will not work for Facebook, Youtube etc. Or at least not well.
Steve