Shut off the power to the home at the existing breaker box / fuse panel. Before attempting any electrical work in a home, the existing power should be disconnected at the source.
Starting with the ground floor, identify each switch, duplex receptacle and light circuit. Draw a map on a tablet that represents each circuit. Installers for knob and tube wiring did not always follow a predictable pattern when running their wiring nearly a century ago. Knowing the scope of your work before pulling the new wire and circuits is very helpful.
Remove one receptacle at a time, and replace the single stranded wires with modern electrical cable. Follow electrical code to determine which gauge wire is used for each circuit. Generally, single lights and lightly used plugs in bedrooms use 14-2, and circuits with more than one light, or more than one receptacle on the same circuit use the heavier 12-2 gauge wiring. Follow and remove the existing wiring all the way back to the fuse panel, and pull new ROMEX® in its place.
Install a new electrical box into the wall for the receptacle, and then mount the new duplex plug in the box by connecting the white wire to the silver screw, the black wire to the gold screw and the uninsulated ground wire to the green screw.
Repeat this process for each plug, switch and light fixture on the ground floor, and then repeat the process for the upper floors if applicable.
Return to the basement to complete the conversion after all the wires are run to the new locations. Call the electrical company to come to your home and remove the electrical meter from the outside of the home. This will disconnect the power to the existing fuse panel so it can be converted to a new breaker panel.
Remove the existing fuse panel. Install the new breaker panel in the same location, and connect the wires which enter the home from the outdoor electrical meter first. The white wires are connected to the hot (or positive) side of the circuit in the breaker panel. The black wires are connected to the grounding lugs in the panel, and the ground wire is also connected to the grounding lug.
Connect each of the new circuits in the home to the appropriate size breaker. The black negative wire is connected to the grounding lug, along with all the bare grounding conductor. The white conductors are each connected to a breaker, and the breaker snaps into the breaker panel.
Have the electrical meter reinstalled, and then turn on the main breaker of the panel. Flip the individual circuits on one at a time, and check the circuits to make sure the lights and plugs work properly.