Remove the baseboard from the wall, as carefully as possible, with a narrow pry-bar and hammer.
Turn off the power, at the service panel, to the room with the circuits you will be replacing. Run an extension cord and work light from an energized circuit into the room being rewired.
Remove all cover plates, outlets and switches from their electrical boxes with a screwdriver.
Test all circuits in the room with a multimeter to ensure the power is off. Set the meter to its 120-volt AC setting. Touch one of the meter’s probes to either the brass or silver screw on the side terminals of an outlet or switch. Touch the meter’s other probe to the remaining silver or brass screw on the outlet or switch. A “0” reading indicates the circuits are off.
Examine the existing wiring, looking up inside the wall from the bottom – from where the baseboard was removed -- with a flashlight. Gently wiggle and pull one end of each wire to see where the wire leads, and what it is connected to.
Run a tape measure over the outside of the wall between the wire’s connection points. Measure and cut new wires for all circuits, with electrical pliers. Consult local codes for wire sizing requirements.
Work downward from the highest switch or outlet on the wall. Insert a fish tape through the opening in the electrical box where the switch or outlet was removed, down to the baseboard opening at the bottom of the wall. Connect the new wire to the end of the fish tape and pull the wire up through wall. Bend the wire around the edge of the electrical box to hold the wire in place.
Lay out the new wire along the baseboard opening to the point where it will be connected to the wire coming from the service panel. If the wire connecting the room’s circuits to the service panel is also being replaced, run a new wire along the route of the old wire, using the fish tape as needed. Strip one inch of insulation from the end of each wire with the electrical pliers. Rough-in all the new wires before beginning to change out any connections.
Remove the existing wires, with a screwdriver, from the connections at the switches and outlets -- one at a time. Connect the new cable, attaching the black hot wire to the brass screw and the white neutral wire to the silver screw. Connect the green grounding wire to the green grounding screw on the outlet or switch. Replace old hardware that does not have grounding screws with new switches and outlets. Wrap each wire tightly around each connecting screw so the wires are tight and do not overlap each other.
Unscrew the wire nuts from any spliced wires with your fingers, taking note of the wiring configuration. For safety, mark each wire with a marking pen, designating each wire A, B, C, etc. Insert the ends of the new wires into the wire nuts and twist the nuts tight with your fingers.
Double-check both ends of each new wire connection, ensuring the wire is securely fastened. Turn the power back on at the service panel. Check each circuit with the multimeter and turn the power back off.
Reinstall each switch, outlet and cover plate with the screwdriver. Replace the baseboard with a hammer and nails. Patch the top of the baseboard with plaster and repaint if necessary.