I live in Newington, and there are no bike paths here.
Your best bet is to go to Farmington and start there, going North.
They have a very well maintained MUT that goes almost to the Mass line, and will supposedly connect to other paths leading to Northampton, Ma. That would be brilliant.
Anyway, here are links that I've used in the past to go riding with my kids (7 and 9 - I'm not real comfortable taking them on extended road trips yet):
I've ridden most of them in the central and Eastern part of the state, and can say they are mostly in great shape.
Here's another link that I've never used before, but it looks pretty robust:
I've ridden the following trails:
*Airline State Park Trail - dirt, very rural, few if any resources. Bring your own water/snacks.
*Charter Oak Greenway - decent shape, a little hilly, several chances to stop and get snacks, drinks, etc...
*Farmington Canal Heritage Trail - good shape, a little crowded with walkers/dogs/babies/kids on nice days. Crosses lots of roads, but well marked and well maintained. Good resources for nourishment.
*Vernon Rails to Trails - dirt, easy to get to, crosses some roads, but most is pretty isolated. Can get a little busy on nice weekends. Not many chances for sustinence.
I prefer to ride on the road, but there are plenty of paved paths in the state, and most are pretty easy to get to. Seems like there are more dirt trails than paved, but either way, they are mostly very well maintained.
I'm new to BikeForums, but not to cycling, and have ridden all over the state of Connecticut and lots of New England. Let me know if you have questions about riding in specific parts of the state.