Presuming that you already have your ticket and a valid ID card (pasport?)...............
Normally, a small and light backpack with clothes only will be no problem to bring as handluggage. But, new rules state that you CAN NOT bring any fluids, nailfiles, scissors, etc in your handluggage. You may want to check on the full list of rules with the agent where you got the ticket.
No worries, airport staff are patient and friendly. They will give you kind instructions on what is expected from you. There are also lots of signs pointing you to the right directions.
When you enter the passengers departure hall, you might get the first check-up on your luggage. Usually they just want you to put it through a scanner, but sometimes you have to take some stuff out for a double-check. This first check-up is not always the case, it depends on the airport that you're flying from.
Then you need to find out on which row is your check-in counter (usually on screen display in the hall, written after your flight number and destination)
Then you go to the check-in counter and line up in the queue for your destination and status (economy, business class, etc), with your ticket and ID-card ready. They want to see those both. They will ask how many pieces of luggage you want to check-in (not counting your handluggage).
When you are checked-in at the counter, they will give you a boarding card. This card tells you what time you must be at your departure gate. It also tells you the number of the departure gate and your seat number.
If you're on an international flight, you now need to go through the customs, the signs will show you where to go. They want to see your passport and your boarding card. They might scan your luggage again.
Once you've passed the customs, you may head to your departure gate. When you get to the departure gate, they will check your luggage again --- very strict. You may have to take everything out. You will also have to walk through a scanner-porch. And often a security officer will perform a personal check-up on you. They will want to see your boarding card. They might want to see your ID card again, to check if this is really your boarding card.
Then they allow you to enter the waiting hall. You sit there until you hear over the speakers that passengers for (your flight number and destination) are requested to board the aircraft.
When you move towards the gate that leads to the aircraft, they will want your boarding card again. They tear off one part and return the other part to you (it has your seat number).
When you enter the aircraft, you look for your seat number and store your hand luggage in the locker above your seat. Or, if possible, under the seat in front of you.
Fasten your seat belt and have a good flight!