You have two choices basically. Either you go to a Travel Agent (ie Flight Centre et al) or you buy them online.
If you choose to go through a travel agent (ie Flight Centre), you give them all the details... for example, the airline you want to fly with, or the dates and/or times that you want to depart or arrive. The agent then "searches" (via computer) for a service that meets your requirements. You then pay the agent... usually by credit card or electronic transfer.
Alternatively, you can go online (either to the airline directly or to a bureau such as flightcentre.com). You then make your selections. You then punch in your credit card details to pay for your "ticket".
Nowadays though, you rarely see physical airline tickets. Most people travel on eTickets. All you get are your boarding pass or "confirmation"; and when you check-in, you present your boarding pass/confirmation and photo ID/passport to get on the plane.
go to cheapoair.com or cheapflights.com or expedia.com, any of those things or you could call the airline directly. you cant travel on a plane by yourself until youre 15 unless you have like a chaperone person from the airport. that costs $50 each way = $100 just for paperwork the whole time. its so stupid, they dont even do anything for you. and youre the first to get on the plane and the last to get off. talk about boring.
The best site for purchasing airline tickets is skyhigh. Best airline for convenient routing also...
Go online and check out thier travel packages!
You buy them