When we travel we take either 2 or 3 suitcases between the 4 of us. Either DH and I share and the DC share another one, or DH and I have one each and the DC share.
If we are going somewhere where we are likely to buy a lot of souvenirs, we sometimes take just 2 suitcases and pack an empty holdall to use as our third piece of luggage on the way home.
We prefer to keep luggage to a minimum so that it is easier to and from the airport and at baggage reclaim. We have medium sized, lightweight suitcases. e did buy big ones, but if we filled them they were over the luggage allowance sometimes so its easier to have medium sized ones and then yo don't get caught out with excess baggage fees.
We all take a rucksack for handluggage, I also take a small handbag, most airlines allow this but you'd need to check yours.
When you get to the airport you would go to the terminal you are flying from, your itinerary should tell you which, and head to departures. You would then queue up for the check in for your flight, all the check in desks have big screens above them and you'll be able to spot your airline logo and the destination, so you will know where to queue.
When you check in they will want any booking details you have and your passports, they will give you your boarding cards (don't lose these, keep them with your passports) and they will weigh and label your luggage. They will usually take your luggage from you there, if you have holdalls or backpacks rather than suitcases you may have to take them to a bag drop area yourself. You will keep your hand luggage.
From check in you will head to security. You will have to take your shoes, belts off, empty pockets. Put everything through the scanners, walk through a scanner yourself and then once you have the all clear then you can head into the departure lounge and look round the shops or have something to eat.
Check your boarding card for which gate your plane will be at and keep an eye on the boards in the departures lounge, they will tell you when you need to go to your gate. Once you get to your gate you can wait until they ask you to board, they may ask for say, rows 1 - 35 to board please, you just wait for your section of the plane to be called, your seat numbers will be on your boarding card. When your section is called, go up to the gate and they wil want to see your boarding card (they will keep this) and your passport. Then you can get on the plane.