There are many different ways to buy tickets, but if you are look to find the cheapest fares you might want to consider one of the price comparison websites. Some of the most popular ones are:
Cheap Flights www.cheapflights.com
If you have flexible travel dates, some of these websites can also show you the cheapest days to travel. Usually, but not always, weekdays are cheaper than weekends. Summer, holidays, and school vacations are often the most expensive as demand for available seats is greater.
I do want to give a word of warning about buying the cheapest tickets. Sometimes these cheap tickets are called “Basic Economy” and have many different restrictions. One of the biggest restriction is you cannot choose your seat. This means you will be assigned any left over seats on the day of departure. These are usually middle seats. If you are traveling with other people you will probably be seated apart. You should avoid these types of tickets if you are traveling with small children.
Also many of the cheapest tickets have restrictions on changing your ticket. If you need to change your travel days you will often have to pay a steep penalty or forfeit the ticket.