Easiest for me, personal travel:
(1) Search a general portal to get an idea of fares, options, etc. Hipmunk, Orbitz, etc. For a domestic U.S. route, I probably only do one search. For an international route or a domestic route in another part of the world, I might run 3-4 searches (different portals) just to see if there are differences between them.
(2) Then, book at the airline site directly.
Corp travel:
(1) Login to our portal. Book ticket.
When I might use a travel agent:
- Specialty trip. For example, golf trip to Vegas or Myrtle...I'll talk to a local TA who plays often and knows what courses are in good shape, where some promotional deals might be, how to get "hard to get" tee times, etc.
- Language barrier. I want someone to book a series of boutique hotels for me in a place where I don't want to try to have broken-English conversations via phone. Local TA in the local language can do better.
- I can't get the airline to accept my credit cards. (EgyptAir being one, Jet Airways being another.) Then I'll use an OTA to ticket and just pray nothing goes wrong with the flights.
Using a travel agent is always a little painful and difficult, but it can be worth it for the right reasons.
Only time I've used a concierge is for something very local: tickets to something, dinner in a restaurant that isn't on Opentable, etc.