Be sure you have purchased the right filter and have all the tools you need before starting. It's a pain to have to run back to the store in the middle of the job, especially since you won't be able to use your car.
It's helpful to have a friend who has done this before around for assistance.
To avoid getting oil all over your arm when removing the drain plug bolt, apply inward force (as if you are trying to push the plug back into the hole) while removing it. When you know that bolt is fully unscrewed, quickly pull it away from the opening. If you're lucky only a few drops will get on your hand. Tie a rag around your wrist when you remove the oil plug. Oil might get all over your hands, but it won't go all the way down your arm, making it a messy clean up. Also, have pumice or citrus hand-cleaner, or consider using disposable gloves to keep your hands perfectly clean. Latex rubber will decay quickly in oil, so nitrile or another material is a better choice.
For a very stubborn filter, using a hammer and a large screwdriver as a sort of "chisel" can push the filter counter-clockwise. Be advised: once you punch hole in this thin filter wall, the engine cannot be started until it's replaced.
There are some oil drain valves on the market that replace your normal drain pan bolt. These can make oil changes much more convenient and can reduce the amount of mess you make.
To avoid spilling too much oil as you remove the filter, wrap a plastic bag around the filter, which will catch any oil that escapes as you remove it. Before disposing of the old filter, it is best to get all possible oil out. This can be done by suspending it upside-down on a heavy wire mesh set across your oil catch basin. As a last step, taking your time (because warm used oil is thin, ie, low viscosity), pour it through a big funnel into gallon jugs. They can be coolant, windshield-washer fluid, anything with a secure screw cap. Leave a few inches of room at the top. Don't allow these containers to sit in sunlight too long, the plastic can get very brittle.
You can lower your cost by buying cheaper oil and filters if you so choose. Remember, though, to always make sure that the oil you use fulfills the car manufacturer's requirements as stated in the manual. Be sure to use the oil specified in the owners manual as to viscosity. That's the number that looks like "10W-30". A lower "W" rating means the oil is formulated to flow better in cold. Below freezing, a 5W oil is preferred. Below 0°F (-18°C), a synthetic 0W oil is worth considering.
Draining your oil with the engine still warm can speed up the draining process. Be careful not to let the hot oil get on your skin