Firstly, relax, there's nothing really complicated about what you are about to do.
Second, relax, there's nothing really dangerous about what you are about to do (honest).
Pack the bag that's going in the hold with as much of your kit as possible (no laptops though, hand carry them always). Lock your bag with an approved lock (try Boots the Chemists for these).
If you have walking / hiking shoes, or shoes with molded soles and uppers, these are preferable as you probably won't be asked to remove them at security. If you can manage without a belt, do so, else take something you can re-thread through your trousers easily.
Take a small rucksack, they're a god-send for all the junk (boarding pass/itinerary/passport/driving license/ipod/phone/keys/wallet/loose change/jacket/sunglasses etc. etc.) you need to take with you on an air trip. Yes, something big enough to squeeze your jacket into, as I find airports are always slightly too warm, or slightly too cold. No liquids! Yeah, you're allowed 100ml blah blah blah, but it just gets complicated.
Don't arrive too early, there's limited facilities on the non-air side of modern airports, but deffo don't be late. Check-in is straight forward, and if you explain it's your first time, they'll be very happy to slow down and detail what's what for you. So, pack your bag yourself, check the weight before you leave home (bathroom or fishing scales), put your bag on the conveyor-belt/scale and hand over your ID and boarding pass. Ask if your bag is "checked right through" to confirm that you don't need to collect it for connections or customs clearances if going international. Smile, this is the stressful bit all done with.
Ask for directions to the Security Check Area, if you can't see it.
On your way along the zig-zag isles to the security check, remove all the bumf from your pockets, your belt, sun glasses and beanie hat and put it all in your small bag. When you reach the end of the queue and it's your turn, if you are carrying a laptop remove it and put it in a tray on its own. Place your bag in the next tray. Only remove your shoes if they tell you too. Smile, and relax, all the staff here are just doing their job. Look through the doorway and aim to make eye contact with one of the staff on the other side, nod and they'll beckon you through and likely not even look at you again. Collect your laptop, and your bag, fetch out your belt and rescue your trousers from around your ankles, else it'll be difficult to get your shoes back on. Remember, you're not in a hurry, take your time (once you've rescued your trousers) and relax.
If you've got lost prior to this, then ask for directions. Don't worry about getting lost, or feeling like you don't know if you're going the right way, airports are designed like this it's an anti-terrorist measure. You've likely only travelled about 50m from the check-in desk as the crow flies. I allow 1 hour from check-in to the gate, that's normally plenty to clear security even on the busiest of days. Realistically it'll take about 15mins to pass through security alone, which is mostly queuing time.
Check-in for your return trip may be possible before you leave, but depends how long you'll be away for. Your hotel can likely get you internet access for check-in purposes, if you ask. Sometimes, when you check the overhead to see what check-in desk / bag-drop for your flight, it'll direct you to a partner airline. There'll deffo be someway to check-in at the airport though, even if it's just at the bag-drop.
I hope you enjoy your flight.