Connecticut colonists, as well as American colonists in general, faced a number of challenges. Early in the history of the colonies, of course, the colonists were simply trying to survive. They basically had to do everything from scratch, starting with constructing shelter, hunting and farming food, and locating drinkable water. Building positive relationships with Native Americans required some time and energy; avoiding positive relationships with these natives frequently led to hostility, violence and even death. Disease was a frequent presence, and in such close colonial quarters, tended to spread quickly. As the colonies drew closer to the mid-1700's, conflict with Britain increased tensions among colonists who became divided over whether or not to support the Crown.
I think one of their more challenging hardships was simply building a colonial economy that allowed them to do more than survive. Connecticut was not blessed with the temperate climate and fertile soil of the Chesapeake and southern colonies and cash crops were never really an option. So the route for these colonists from subsistence economy to anything like a sustainable, diverse one was a long one that took more than a century.