When you get to the airport you will need to go to ticketing. This is where the big signs say "domestic departures"
If you have to check luggage you have two options (usually) you can use a sky cap, and they will do it on the curb outside for you, and they will issue you your ticket. If you do it outside you will have to tip the sky cap around 2 dollars per bag.-this service isn't available at all airports or with all airlines.
You can also go to ticketing inside (this is what 90% of folks do). You will need to locate your airline, and get in their line. There are always agents there to answer questions about what lines you need to be in. Then you will need to show your id and usually a credit card, or frequent flyer number. They will tag your bag, and you can proceed to security.
For security you will need to show the agent your boarding pass and your id. You will then go through the line. If you have a laptop you will need to remove it. You also can't have liquids that exceed 2 oz. If you do have liquids that are travel sized they need to be together in a plastc bag (or one that is see through) They give you bins to put your belongings in. They will want you to take off your shoes and coat and put them through the scanner.
You will then walk through the scanner, if you beep they will help you figure out why you are beeping. They might pull you aside for additional screening. This is random, and nothing to be concerned about.
The security agents can appear harsh at times, but they are just trying to be efficient, don't let them upset you or intimidate you, if you have questions just ask them.
Then you will need to find your gate and wait for your flight. It isn't a terrible experience at all.