In a HCOL, the things I have found to be the most difficult to deal with are:
1. Housing Costs.
2. Costs for any type of personal service, but especially things like daycare, preschool, summer camps, etc. Also for skilled labor (e.g. electrician), though for that sometimes you will get people who live 1.5 hours away and drive in, and they bump up their rates to account for commuting time.
3. Higher taxes.
Costs for things like groceries, amazon junk, gas, etc. are all on par with national averages. (Basically, anything trucked or piped in from elsewhere is the same, or sometimes even cheaper if you are in the suburbs because there is an enormous variety and so more competition than a smaller city or rural area, without quite as much challenge in trucking it in and out than, say, downtown Manhattan.) Healthcare is also the same, I think just because healthcare workers like living in big cities so they charge a premium to live other places. In my particular county I also need to deal with like a million speeding cameras (I think my jurisdiction gives out like 50% of the camera tickets in the country or something absurd like that) and high cost heavily regulated alcohol, but that's just my own little cross to bear and not as much of an issue in SE Connecticut. But I think you have higher property taxes so that's actually more material.
You can defray #1 by buying a smaller house and #2 by just purchasing fewer personal services (doing more things yourself).
HCOLs frequently have some benefits, such as good public transit, good public libraries, good public parks, well funded schools, free festivals and parades, free museum tickets available from the local library if you book in advance, etc. So there are some corresponding benefits, though not enough IMO to outweigh the costs, at least for a family. Living to me seems easiest in popular "secondary cities" with good economies (e.g., Columbus, Ohio). I live where I do mostly for family and partly for maybe misguided nostalgia (I grew up here, so it's just home) and partly because I'm here now and moving with kids is hard. But I don't actively recommend it.