To be secure you have to ensure that all your traffic (including your DNS requests) goes through the VPN. This should be the default configuration for such anonymization VPN services.
In this case the ISP would see just the VPN connection (and sizes of the packets, their timing etc.). From the DNS requests going outside of the VPN they would easily see the addresses you are connecting to!
Dangers of anonymization VPNs
The problem on anonymization VPNs is that the VPN provider can easily see (and modify!) all your traffic! In many cases using anonymization VPNs (without using end-to-end encryption like SSL/TLS) could be very dangerous!
The VPN providers can also log the traffic and provide the logs to others (government institutions etc.).
More secure alternative - Tor
If you do not want to use a high bandwidth (e.g. downloads) and low latency (e.g. VoIP), much more secure alternative could be the Tor network. To get the basic idea you can see How should one explain Tor? See also the home page with more information: https://www.torproject.org/
One of the main differences is that no-one sees your IP address and your open communication at the same time so when using Tor properly practically no-one should be able to connect your IP address with your open communication. There is a high number of exit nodes (from which your traffic goes to the destination computers) which are selected randomly. The routing path randomly changes every 10 minutes.
Avoiding detection of Tor
From some characteristics ISP can detect that you are using Tor. There are modules which can encapsulate the traffic so that it looks like a regular SSL/TLS and be almost indistinguishable from HTTPS traffic. See How are 'Pluggable Transport' bundles different from 'regular' Tor Browser Bundles?