Mark, I'm a new diver, too.
I've only been diving in New England for less than a year.
I'm sure there are some good beach diving spots in Connecticut that I don't even know about, so hopefully some others will post some helpful information on beach diving spots in Connecticut.
I'm sure you're aware that there are also some good lake diving options in Connecticut.
But in response to your question, here are the beach diving options I'm familiar with for Connecticut divers.
I've been to Stonington on the Connecticut shore. From West Hartford, I'd plan on a solid hour drive to the dive area, maybe a little more. I found it to be a mediocre beach dive, at best.
I've also been to the Rhode Island beach, in Jamestown, at a site known as Ft. Wetherill. I know it's not in Connecticut, but it's probably about a 1 1/2 hour drive from West Hartford. I think the extra driving beyond Stonington is worth it.
Again, I know it's not Connecticut, but Cape Ann in Massachusetts is a good beach diving destination. From West Hartford, it's a solid two hour drive. There are numerous beach diving options in Cape Ann and they are typically far far superior to the diving at Ft. Wetherill or Stonington. I've made a few of weekend day trips up there recently and it was well worth it.
Hopefully, if you post specifically where and when you'd like to dive, and that you need a buddy, you will receive some responses from the folks on this Board.
Here are a couple helpful websites with information on local diving options:
http://www.mit.edu/activities/scuba-club/Dive_Sites.html
http://www.scuba-newengland.com/DiveSites.asp
http://www.divenewengland.com/
I hope some of this helps. Good luck.
Michael