Navigate your Internet browser to a multiple airline search engine such as Kayak.com or Momondo.com. Be sure to investigate multiple search engines before you purchase your ticket, as each has different strengths. Some are excellent for domestic flights, others provide superior service for international flights.
Select the arrival and departure cities, dates of departure and return and click on the search flights icon. Typically, a "flexible dates" search option is also available.
The search results will display available airfares for the information you selected. You can filter your results based on personal preferences such as number of stops, airline and price.
If possible, change the parameters of your search to optimize your ticket for price, airline, etc. For example, flying on a Monday as opposed to a Sunday may decrease the price of your ticket.
Once you have selected a ticket, the search engine will redirect your browser to the website where you may purchase the ticket. You will be asked to enter personal information and form of payment. Be careful, many sites offer additional promotions that you may not want to purchase. Often, you must opt-out of these options.