If you setup your VPN client correctly, making sure you don't have DNS leaks, and the VPN isn't using a split tunnel, all your ISP will be able to see is encrypted packets with a source (your IP) and destination (VPN server). The VPN server then decrypts these packets, and sends DNS requests to it's DNS servers over it's own ISP. The bits of data that say (destination: news.giganews.com) are encrypted, until they reach the VPN server, by this time they are out of reach of your ISP.
When the data packets need to be sent back to your IP, the VPN server encrypts them, and sends them over the secure tunnel with the source (VPN server) and destination (your IP).
Here is a fancy diagram to better illustrate how a VPN works.
As dnoyeB sayed, if a government, or other authority wants to tap the connection after then VPN, or the VPN server itself, then your privacy is compromised. That is why it is extremely important to research the privacy policies, reviews, and reputation of a VPN provider before deciding to use them.