I firmly believe that everybody who owns a car should know how to change their own oil. Now whether or not you choose to or are physically able to is one thing, but again, I think it is a skill every car owner should possess.
I actually enjoy changing my own oil, there something almost cathartic about it. Getting dirty (and you will get dirty) to help my car to run better, having a chance to take a closer look at a machine that most people generally don't understand. I think to think it helps me stay more in tune with my car. Knowing what is normal on your car and what isn't can save you a lot of money.
For example, one day driving to work my check oil light turned on and off a couple times. I know that is strange, so I made a mental note to check my oil level after work. When I drove home, the same thing happened (again, strange). I popped the hood, checked the oil, and lo and behold it was almost empty. Investigating under my car I found I hadn't tightened my oil filter enough and had been dripping oil for a couple thousand miles... Now since I found it the day my car sensed something was wrong, it was only a $35 mistake and required a friend to drive me to Autozone. If I had kept driving on it I could have done some Serious damage to my engine and may have even been driving a new car by now.
Anecdote aside, here are the easy steps to changing your oil:
1) Remove your oil drain plug (the oil will drain faster if it is hot so maybe run to the auto parts store first).
2) Replace drain plug after the oil has drained, making sure the plug is wiped clean. If your car has a crush washer, you will need to replace that as well (another mistake I have made in the past...) and they're usually cheap - mine are about $5 for 3.
3) Remove the old oil filter.
4) Wipe some old oil on the new filter's gasket and screw it into place. You don't need to crank it as tight as possible, just make sure it's tight.
5) Add fresh oil until full, using the dip stick to confirm a couple times. I generally fill 4 quarts and then turn the engine on for a few seconds before checking.
I highly recommend having plenty of automotive paper towels at the ready as well as rubber gloves (which I clearly did not have with me this time). If you don't have any rubber gloves, you should probably get some hand cleaner - there are plenty of options at your local automotive store. You can actually even get oil change kits, usually for less than $30 that come with a funnel, oil catch pan,
I also take the time to change my engine air filter, cabin air filter, and check my tire pressure whenever I'm changing my oil.