Begin by positioning the vehicle in a safe area, hopefully far from any traffic, try to find a place which is flat and has plenty of space to work in.

Gather the tools needed such as a jack, wheel-nut spanner and the replacement wheel and tire.

Take a large hard to move or roll over object (brick, log, etc.) and place it behind one of the tires not being removed. 

Locate and position the jacking points under your car, there will be one for each wheel, position your jack directly below.

Start winding or pumping the jack until it just touches the bottom of the car.

Check to make sure the position is correct, then carry on SLOWLY.

Stop jacking after the car's weight has been reduced from the tire, but not at the point where the tire is off the ground.

Remove the hub-cap if your car has them.

Take the wheel-nut spanner and loosen each nut on the wheel for 1 - 1.5 turns of the spanner.

The goal is to loosen them slightly, and not take them off at this point.

With the wheel nuts loosened, begin jacking the car up until the wheel is now clearly 1 inch or more off the ground.

Use the spanner and REMOVE the nuts from the wheel.

I advise removal of the wheel nuts in a order which begins with one and then continues to the opposite nut, and then alternates as best as possible.  Do not just go around in a circle.

When loosening and tightening the nuts, arrange the spanner so that you are pressing down (with gravity). This will remove risk of injury to your back and also allow you to use your body weight rather than just your arm strength.

Now, remove the old wheel and place the new or spare wheel and tire back on.

Replacement of the wheel or lug nuts is next and should begin with each hand tightened as far as you can easily get them on.

Then take your wheel-nut spanner and alternating with opposite positions on the wheel tighten each as much as you can.

The idea here is ofcourse not to allow them to fall off, or the wheel will most likely follow suit.

Once satisfied with that the wheel and tire are safely secured, begin to wind the jack down or release the hydraulic valve so that you can remove it from underneath the car.

If you are replacing more than one wheel, it may be of use to buy some Axle stands so you're not taking the car up and down all the time.

Be careful when jacking up your car as they are very heavy, and jacks have been known to fail. You don't want to be crushed underneath it.

It is advisable to get a garage to check the tightness of your wheel nuts with a calibrated tool so that they don't fall off when you are driving.