gehenna: Can you detail exactly what equipment you have? My reading of your OP is that you have an Ethernet Dongle for your MacBook Air (since the Air doesn't have built-in Ethernet)
If that's the case you should simply have to plug the dongle into your Air, then plug the ethernet cable into the dongle, and then the other end of the ethernet cable into your router.
If you aren't able to connect to the internet once all that is done you might not have your Air set to automatically pick up an IP address. Check this in System Preferences > Network.
Click on the Ethernet icon on the left panel of the window and it will show you the settings that are applied on the right. You want to set this to Automatically Detect Settings, or wording similar to that, and then click Apply.
Once you have established the internet connection successfully you need to set up the VPN connection to work.
You need the details of their VPN "server" for this. If it's standard Cisco IPSEC VPN that most businesses use then this is built into the Mac so you can add the connection in the same System Preferences area - but instead of clicking on Ethernet, click on the Plus + sign at the bottom of the window and choose VPN. Once this appears you'll see a number of fields on the right that you have to populate with the company settings (IP Address or Hostname, username, password) and then an Authentication Settings button that brings you to a window where you put in the Shared Secret Key and a Group Name.
If it is some different type of VPN, maybe SSL or a different vendor like CheckPoint, then you'll need to download or be given specific software for this to work as it won't work on the Mac without installing 3rd party software.
If all this is beyond you then you should take the Air to your IT people and get them to set this all up. It's ridiculous that they would give you equipment to use that isn't already configured. And if they have already configured it then they should give you instruction on how to use it.
If you do end up with a working WiFi connection you can leave out the Ethernet dongle steps and just use WiFi, and then add the VPN connection as detailed.