Acoustic Insulation

That's a bit crap!! Some houses are just built to a budget and small cheap fixes aren't included for this type of issue.

There are two types of noises, conversation vs impact. Not much you can do about impact from a retrofit point of view, if you're semi detached, a door banging next door will be "felt" throughout the structure unless it's designed out. Some buildings are worse than others.

Usually to lower noise next door you'll need two layers of plasterboard with no overlapping joints, each offset to the other. So each layer starts 50% offset to the other for instance. Ideally you use a resilient channel which is a metal frame you attach to the wall first and then fix the plasterboard layers to it - this further improves the plasterboard's noise performance. Depends if you have fitted wardrobes and access to the dividing wall fully though?

As you own your own 1st floor, you can take up the floorboards and fill between the joists with rockwool - note electrical paths though, get professional advice re: uprating cables or protecting them from overheating with some trunking, not my area but you don't want heavily used mains cable being surrounded by insulation and overheating. Plus any downlights should be fire rated if below a bedroom which it may not be. That will help vertical noise transmission at least.

There are specialist suppliers who have more serious stuff - I used some in my master bedroom on the wall and bought heavy duty lining for a new laminate floor upstairs. That's worked great except at the very wall edge where my flatmate works below me right there, and I hear him on the phone through there. TV/Music etc is fine though. I'll PM you the guy I bought those products off but rockwool / acoustic plasterboard and that resilient channel should be enough.