It sounds like you will need to use a firewall and a switch.
The firewall will allow you to control access to the internet from and to your network while the switch gives you more ports to connect to workstations or devices.
Your internet connection will plug into the firewall via a CAT5 or CAT6 cable. This is where you can configure rules as to what users can access or can not access. Check out the barracuda spyware firewall as a good example. This will make your jobe easier as it has predefined rules for blocking webmail and social networking sites. If not you can get a Cisco or Sonicwall and try and sort it out yourself with their technical support.
Onec you have that setup properly you will then have a switch connect to the firewal via ethernet cable to allow for the 10-20 computers to be connected.
One cable will need to be run from each network jack back to a punch panel and the punch panel connects to the switch. The individual computers will be connected to the network jack and the entire network.
You can also use software firewalls on the local machines to try and block the users from accessing their webmail but it can easily be over come.
If you need further assistance please let me know.