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 todays 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.
Its 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 dont 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....