This is a free way to go between terminals 1,2,3, and 4 or 1,2,3, and 5. Be forewarned: Although well-managed and on schedule, the trains are not particularly frequent. Wait times can be up to 15 mins
To use the Wi-Fi, if you have configured specific DNS servers, such as Google's, you need to remove them from your config and use the DHCP-supplied one to access the captive portal and log in.
Best option for a speedy trip in to London Paddington, 22 mins, it is pricey though and if you are not going to West London or if there is more than one of you then consider a cab