Sketch out a plan of the basement, and mark the locations where you want to install lighting fixtures, light switches and electrical outlets. Follow the rule of thumb that no more than 12 electric outlets or lights should be placed on a single circuit. Also check your local building codes for any relevant information and rules for your area. If in doubt, consult with an electrician.
Place your outlet boxes, switch boxes and lighting fixture boxes in the locations you've chosen. Nail each box to the adjacent stud or joist to secure it in place. Remember to take into account the width of the wall or ceiling material when placing the boxes. For example, when placing an outlet box, the front edge should extend 1/2 inch in front of the stud if the wall material is 1/2-inch thick. Place outlet boxes with the bottom edge 12 inches off the floor, and switch boxes with the bottom edge 42 inches above the floor.
Run your wire according to the plan you've sketched out. Don't connect any wires to the circuit breaker panel yet -- you'll save that for the end. Measure the lengths of wire you'll need to reach from each point to the next and 3 to 4 inches onto each end. When routing wire through studs, drill holes in the studs at least 1 1/4 inches back from the front of the stud and pull the wires through. Secure wires along the length of the studs with cable staples. Be sure not the puncture the cover of the cable with the points of the staples. Route the ends of the wires through the holes in the outlet boxes.
Trim back the ends of all the routed wires, and remove 1/2 inch of insulation from the black and white wires of the cables. Secure the wires to the screws of the switches, outlets and fixtures by attaching the white screw to the silver screw and the black wire to the brass screw, using a screwdriver.
Place the outlets and switches into the boxes and attach with the mounting screws.
Turn off the power to the main circuit breaker panel to the home. Remove the cover of the panel. Attach the wires leading to the basement circuit to the panel and add the new circuit breaker. Replace the cover and turn the circuit on.