Living in Connecticut, particularly closer to Hartford, isn't like what most people describe when they talk about East Coast living (i.e. New York and Boston). We've still got relatively affordable homes, especially relative to salaries at the health center.

Hartford itself is small. If you're coming from Nebraska and doing medical, I'm guessing you're coming from Omaha and the medical center there - it will seem similar in scale. Although Hartford is 1/4 the population of Omaha, the Hartford area has about 50% more people.

CT has a lot more rural parts than people think. Especially in Fairfield and Windham counties. It's also a lot hillier than people think coming from outside of the state. This leads to some great outdoors activities.

All that being said, figure out where to live first. Given that you mentioned graduating recently and working at the health center, this probably means West Hartford for you. Which is great. It's got a nice downtown, some good food, and is a little more upscale. You can choose to live in Hartford, but the additional commute would be annoying. Other towns in the area such as Farmington and Avon are going to be more expensive and cater primarily to large single family homes. From there, start looking for things to do and see.