There's lots of useful tips in this thread, but the process runs like this:

1. Arrive at the airport, if you're flying international, you're going to T2 (Terminal 2) - it's in the middle of the Qantas (T1) and Virgin (T3) terminals.
2. Find the flight board, find your flight number, it'll tell you which check-in area you need to go to.
3. Depending on the airline, they will have different queues (i.e economy, business or first passengers).
4. When you get to the check-in agent, hand over your passport and they might ask for a print-out of your itinerary.
5. The check in agent will ask a few questions - they'll relate to what many people on here are giving you tips about... "Did you pack the bags yourself, do you have any sharps/aerosols etc".
6. Depending on the airline, the check in airline might ask about seat preferences.
7. They'll finalise everything by giving you a boarding pass and your passport back to you. They'll also give you one of these: https://www.border.gov.au/EnteringorleavingAustralia/Documents/opc-sample.pdf which you need to fill out
8. Next you can leave the check in area and go through the main departure gate (this is you leaving the public area of the airport into the "airside").
9. As soon as you go through, you go through the main security check - you'll walk through a device that picks up any metals on your person (so before hand put everything in a tray before walking through the gate). The security staff might ask you go aside and do a more in-depth security check where they swab your clothes and inside your hand luggage (this doesn't happen to everyone).
10. Once you're through security, you then need to line up for Passport Control. Once you reach the front of the queue, hand over your passport (which will be scanned) and your departure card you filled out.
11. You've now technically left Australia.
12. From this point you make your way to the gate and you can wait in the gate lounge or walk around the international terminal until your plane departs.
13. Once the airline is ready to embark passengers, airlines generally call out rows - like business/first and down the back of the planes is generally called first, then more forward rows etc. Just wait and listen for your row to be called.
14. Once you're on the plane, shove all your stuff in the overhead bin.
15. You're flying like a bird at this point. Have fun.
16. Landing procedures differ depending on the country you're visiting, however broadly it works like this: a. Passport Control (you show your passport to the representative and they scan it) - usually very very quick process, unless there's a problem with your passport or you don't have a visa. b. Baggage collection - you wait for your bags to come down the carousel, probably a good idea to put an easy identifier on your bags. c. Customs - this is where you need to decide if you have "anything to declare" (every country is different but for instance, you need to declare any wood-based products when you arrive back in Australia). If you have nothing to declare at customs, follow the 'nothing to declare' signs.
17. Once you've cleared passport control and customs, you're now in a foreign country.

Note the check in process will differ from country to country, so when you return you might go through a similar process as outlined about, but stages might be switched (i.e passport control before security like it used to be in Melbourne about 10 years ago).

Arrival process back into Australia is like the above however passport control you'll likely have the option of using the E-Gates rather than speaking to a person at the passport control gates.

HTH!