+1 for ecat026’s recommendation of GFCI outlets. This was first suggested to me by the home inspector to bring my receptacles to code. My home was built in 1910.
A GFCI protects you by severing the flow of power whenever it detects ground faults or leakage currents. They are easy to install (do your homework—shut off the breakers before work, etc).
I swapped all receptacles before putting my computer and hifi equipment on them, and I have not had any trouble. Over time you can look to run dedicated power from dedicated breakers to dedicated hifi outlets and keep building up, without replacing your whole grid.