For those traveling within the United States, it is a good idea to purchase the tickets as soon as possible. Book your tickets as soon as you can confirm your travel dates. The earlier you can book your seat, the less expensive the ticket will probably cost.
If you are planning a trip to other countries, it is wise to first look at what kind of visa and passport requirements you might need. All countries require a passport for admittance, so the proper passport should be obtained. The real issue is to find out if you need a prearranged visa.
Most countries will grant a visa--if it is required--upon entering the country, but for others, you will need to apply for one. For example, if you plan to travel to Iran, you will need to apply for a visa through an Iranian embassy. This can take weeks if not months, and even after the wait you are not guaranteed to receive the visa. Because of this it is a good idea to wait and purchase the plane ticket after you know you'll be able to enter the country without any problem.
For those that don't have a preference on where they travel, it's probably best to wait to book your flights until a few days or weeks before you plan to go. If a plane hasn't been completely booked, the airline will normally offer discounts on the remaining seats. This can be the perfect opportunity for those that want to travel but don't want go over budget doing so.