As everyones said, start with all the connections. Maybe even replacing all the receptacles as they're probably very old and worn out (commercial or spec grade is best for high amp draw things like vacuums etc)
I'd approach this with extreme caution and expect anything and everything to go wrong, old wiring is almost certain to have been messed with, butchered and added onto over the years and once you start tearing into it you may be in for quite a few surprises.
I have done my share of rewiring many things, including doing my grandmothers house (from 1950) after the floods of 2008, her wiring appeared in great condition but once I tore into it I found MANY scary things that had been done over the years... and her home was lovingly cared for over it's life!
And yeah we don't have anything but conduit here! Older homes in the suburbs had knob and tube but never in the city of Chicago. My dad owns a commercial/residential building built in 1915 and it had gas AND electric lighting installed when it was built, they had combo fixtures all over the place that had both gas & electric lamps. (the light switches were the old round knobs that you twisted) The third floor is the only floor that still has it's original wiring and it originally had just two circuits serving a 1,300 sqft apartment and the circuits were set up in a staggered configuration so the very back room and very front room are on the same circuit and vice versa with other rooms on the second circuit.
Over the years an A/C circuit was added, as well as one for the kitchen outlets and one for the washer and dryer. We did go through and replace ALL of the wall outlets and switches as I tend to be VERY careful when dealing with old electric and do know that electrical fires often start with poor/failed connections.