I love no brainers!

You can insulate in any one of several acceptable ways and save money on heating, cut nasty drafts and reduce outside noise penetration.

For greatest noise reduction and since "cost [is] not really an issue," use cc spray foam on the walls and rim joist areas, followed by drywall. If sound reduction from noisy basement activities is your prime aim, insulation is a minor factor in reducing it. Hang a ceiling or two on resilient channel or use a sound attenuating wallboard, such as Quiet Rock and don't use recessed lighting of any sort.

Here is a link that might be useful: Basement Insulation

The only thing that works is sprayed closed-cell polyurethane foam insulation covered with mold-resistant drywall or cement backer board. It's expensive but difficult design problems always require more expensive materials. If you want simple solutions it is best to build above ground.

You have to make a decision on were the building envelope will be for heating and cooling.

If the basement contains plumbing lines it needs to be inside the envelope.

Putting insulation between areas inside the envelope has limited value.

"If the basement contains plumbing lines it needs to be inside the envelope. " That depends on your climate. This is essential in about 1/3 of the country, marginal in 1/3, and not necessary at all in the other 1/3.

OP is in the nutmeg state.

Last time i checked it freezes in CT.