Just fifteen years ago there were only two places you could buy airline tickets. At an airport or at a travel agency. You could do this in person or by phone but the results were usually the same. In today's world, you still have those two options but now you can also buy airline tickets from the comfort of your own home by using the Internet.
The airport is a no brainer. Of course they will be more than happy to sell you airline tickets. After all, their jobs depend on it. The problem with buying airline tickets at the airport is that it can be a royal pain to park and wander your way around until you reach the airline you wanted to get the tickets from. It's just not worth the hassle unless you live in a smaller town and your airport only has a couple of gates. You still might have to pay for parking though.
Travel agencies, though not as popular as they used to be, still exist here and there. Travel agents are more than happy to do the job for you. You don't have to pay them extra either. Most travel agents that are still in business have had very long relationships with various airlines and might be able to get you some great deals. Don't overlook this option.
The Internet is the easiest and most popular way to find airline tickets. Thousands of websites exist that only have one goal. Get you to buy tickets from their site. This lets them compete with one another and you are certain to get a good deal from one of them. Buying tickets online also lets you change your departure dates and location so you can try to get a better deal.