While searching for cheap international flights, I found a site at
www.airvalues.com where I can book my student international flights
online. They state they have been selling airfares online for over 10
years. How can I verify the viability of such companies and trust my
bookings will be made and paid to the airlines?

Where do you want to go to and from? I think that might help people to
answer this question.  I don't know anything about airvalues.com,
though it looks like they are a legitimate travel agent.  However, a
charter airline such as Zoomair, Ryanair or Easyjet may be even
cheaper depending on where you would like to fly.