In some jurisdictions, a qualified electrician must install new electrical wiring. Even if that isn't a requirement where you live, remember that incorrect wiring could cause a tragic fire. If you aren't totally confident in your ability to install the wiring properly, have an electrician do it. Your family's safety is well worth the price.
For your own safety, obtain an electrical wiring permit before beginning the job. Have your wiring inspected and approved before you close the walls. Because electrical codes vary across the country, the following information is quite generic. Check your local codes before implementing any of the following suggestions.
Household wiring is generally 14/2 or 14/3 NM plastic coated (non-metallic sheath) wire. It is sometimes called by its trade name, Romex. The numbers 2 and 3 indicate the number of carrier wires inside the sheath. 14/2 will have 2 wires (one black and one white) plus a ground wire (bare copper). 14/3 has three wires (black, white and red) plus the ground wire. 14/2 is used for wiring outlets, lights, and switches. 14/3, with its extra wire, is necessary for wiring 3-way switches.
Drill .75-inch holes through the 2x4-inch framing lumber 15-inches from the floor to run your wire. Install the boxes for your outlets and switches, then run wire to them. Start close to your electrical panel and pull the wire through the framing.
At each outlet and fixture location, loop 12-inches of wire. Nail a metal protector plate to the stud over the hole protecting the wire to avoid driving a nail into it after the wall is finished. Each circuit can have a maximum of 12 outlets (not counting switches) on it, but having more circuits with fewer outlets is a good idea, since you are less likely to overload a circuit. It also allows for future changes and growth. Install any coaxial or telecom cables.
Fluorescent lights provide great overall lighting in a finished basement, while recessed lights are very popular for ambient lighting they provide. Installing different types of lights on different circuits will provide flexibility in how you light your room for different activities. Keep in mind that many basements have small windows and don't get much natural light, so installing lots of lighting will make your basement brighter and more inviting.
Building codes generally state that the wiring for a light fixture cannot directly connect to a fixture. The connection must be in a junction box using approved connectors (wire nuts). Some recessed lights are manufactured with junction boxes as part of the fixture. With others, you will need to install a junction box close to the light fixture itself.
The wire running from the junction box to the fixture itself should be 4-inches long to allow you to remove the fixture. It must also be in an armored cable for extra strength.
Installing 3-way switches at the top and bottom of the stairs provides convenience and lets you turn off the lights once you get upstairs. Having room lights controlled by switches at the bottom of the stairs will allow you to light the room as you enter it, which is much more convenient than having to walk across a semi dark room to reach a light switch. If there is a switch for the furnace, it should be located outside of the furnace room, preferably near the top of the stairs.