Here we go. Simple article, simple job.
First off you will need 5 quarts of oil, and an oil filter.

What is recommended to use is 10W-30. I am using 5W-30, because it is winter. I will switch back to the thicker oil when it gets hot out again. For an in-depth article on motor oil, see here:
http://www.boss302.com/oil.htmNow, jack your car up so you can get under it, or put it on a curb, like I do.

Simple and effective. I am using a 1992 Mustang for this article. You can use the information here to change the oil on pretty much all 5.0L Mustangs, from 1979 onward, though there are likely a few minor differences.
Now pop your hood, and remove the oil cap.

Now it's time to get some parts. Here's a photo of everything you will need.

We've got the oil, oil filter, oil funnel, 3/8ths 6-point sockets, 3/8ths ratchet, two oil filter wrenches (more on that later), a box 'O rags, and the drain pan.
5.0 Mustangs use FL-1A Motorcraft oil filters, and that's true for all Fox-bodied Mustangs at least. So, if you have a 1979 Cobra 5.0, you can use this filter.
One note on the sockets that I used. I highly recommend that you do NOT use 12-point sockets, as they can strip bolts, and that is never a good thing. Use 6-point, and only 6-pt., for oil changes. Your car will thank you.
On to the hard part.
Get your drain pan, rags, ratchet, and sockets, and crawl under your car.
You will first see the front oil plug here:

Ignore it for now. I like to change the oil from the back to the front, first the rear plug, then the front one, and then the oil filter. This style makes it so you don't have to move the drain pan any more than is necessary.
If you look upwards, you will see the oil filter.

Ignore this for now as well, as we will be getting to that last. We also will be changing it from the top, not the bottom.

Now we see the rear plug, in all its (very dirty) glory.
For my Mustang, both of the oil pan plugs use a 5/8ths socket. Yours is the same, unless an aftermarket oil pan was installed on your car at some point, or some idiot stripped the threads on one of the plugs, and installed an oversize one. That is the case for my '90 Mustang, which is one of the reasons why I am not using that car for this article.
First, put the drain pan underneath the rear plug. Now loosen the rear plug, until oil is dripping from it.

Unscrew the plug with your hand, and be careful of hot oil. Make sure you have a firm grip on it, so you don't drop it in the drain pan. It's no fun looking for a plug in hot oil, trust me.
Let the oil drain out, like so.

Wait until there is only a slow drip, and then put the plug back on, and be very careful not to cross-thread it! If it gets stuck, unscrew it and try again. You do not want to strip the threads on the oil pan, as it costs big $$$ to get the pan pulled, and re-thread the hole.
This is one of the reasons why the pan is steel, and the plug is aluminum. The softer metal of the plug will strip before the pan will. Its a saftey precaution. Once it is hand-tight, give it a little "bump" with the ratchet. Do NOT use Arnold muscles here. It only needs a good nudge to stay tight, about 5-10 ft-lbs of force, the same as needed for spark plugs.
Go to the front plug, and repeat the process. Afterwards, you will need to do a bit of cleaning:

Here's where the rags come in handy. Use them liberally. A clean ship is a good ship, as they say (in the Navy). I also like to clean the oilpan itself of any oil and other nastyness (dirt, grime, etc.) while I'm down there. It makes things look much better, though nobody will know about that but you.
Do the same for the front plug as you did for the rear plug, and put it on carefully.
After that, position the drain pan below the oil filter, and get out from under the car.
I use a "special" oil filter wrench for my Mustang, as I use Motorcraft oil filters. The universal band-type works just as well if you are using an aftermarket oil filter.

Now, just reach into the engine bay, and loosen the old oil filter with your wrench (Remember: righty tighty, lefty loosy). Once that is done, carefully unscrew it by hand, and make sure to have a firm grip on it so it doesn't fall into the (now full) oil drain pan. Once it is off, you can either carefully lift it out with it still full of oil, and drain it into the drain pan outside of the car, or simply upend it now, and dump the contents.

Once the old oil filter is off off, you will see this:

Get a rag and clean up the threads and the surrounding area. Pay attention to the raised metal circle too. That's where the rubber gasket on the oil fiter seals the oil in, and prevents a leak.
Before you put the new oil filter on, there is some prep work you shouild do. Guess which is the old filter, and which is the new one!

On the new one, you should put a liberal amount of new oil on the rubber o-ring, to help seal it to the block.

After that is done, hand-thread the new filter onto the engine, and put it on finger-tight. After that, give it a "nudge" with the filter wrench, and tighten it on about a 1/8 to 1/4 turn more. A nudge. Nothing more. If you want to do things exactly "by the book", you can go ahead and only put it on hand-tight. Both methods work just fine.
Now it is time for the fun part: pouring in the new oil.
Get another rag, and the oil funnel, and set them up like so.

The rag is to keep any oil from going where it doesn't belong. Just pour in all five quarts slowly, so none spills. After you're done, just put the oil cap back onto the spout on the valvecover, and you're set. Cleanup is all that's left. Put the used oil in a sealable can or jug of some sort (I use empty orange juice jugs), and take it to your local disposal facility. If you don't know where that is, call the police (use 411 if you don't know their number, NOT 911. 911 is for emergencys ONLY). and ask them. They should know.
Also, have fun with you're newly rejuvenated Mustang!