I live within 1 mile of the Cheshire - Hamden portion of the East Coast Greenway. With a little planning on some quiet back roads, you can ride a nice quiet route all the way from Hamden to the MA state border. As others have mentioned, you can get to the northern portion (Avon to Granby) driving down Farmington Ave.
With that said, even though I am close to the path, it is a disaster to ride it anytime there are people on it. There is a 10 mph limit, mostly ignored, and multiple road crossings to slow down and navigate. It is fine for a quiet pedal with the kids, but anything more it is out of the question. I use the path to avoid riding over a section of road that has no shoulder and limited sight distances.
You are living in a state that has some of the nicest bicycling routes around - on regular roadways. Scenic, hills if you want them, flats if you'd prefer (just stay in the river valleys going mostly North to South), not a lot of cars depending on the route etc. etc. Heck, even if you live right *in* Hartford, it is only a 10 minute drive to a commuter lot if West Hartford, where you can kick off any number of suburban, car-free rides. North and West of Hartford are particularly rural. If you check out the local bicycle shops I bet they can share a number of preferred routes.