If the health center you're referring to is UCONN health, then I'd check out West Hartford center (aka blue back square), it's literally just down route 4 / Farmington Ave from there. There's a lot of restaurants and it has a pretty lively bar scene (at least pre-pandemic).

Also my favorite hidden(ish) gem in town is a Chinese restaurant called butterfly that does karaoke on Fridays and Saturdays. The food is good too.