I don't think you can just 'Show Up' at Lime Rock and get on the track... You need to be part of a 'Club' or sponsored event.
But CT does have an
AutoCross Club/Team
They charge a membership fee, and then they host a number of events, both on track and in parking lots with cones you can go to. I thought I remembered it to be ~$200, but it looks to be much cheaper now.... You'll find details on the site
__________________
1996 Z-28: Bolt-Ons, Harwood Cowl Hood, 18" Black Z06s
2005 Subaru Legacy GT - Stage 2 - My DD
2012 Audi S4 - The Wifee's Ride (Wait - the wife drives the fastest of the stable?

)