Google. For your major airports, or if you know about where you're routing, try the airlines that first spring to mind. AA, UA, BA, QF, etc. You'll often find good deals at the airline websites.

I then take a trip to the webpage of the airport. Often they will have a little pearl of information for you (eg there may be a European airline that only flies in at 3am every third tuesday that is cheap as anything)

Then check out your orbitz, itn.net, cheaptickets, etc. Or as above, google. Something like "LAX +bhm site:flyertalk.com" as your search string [without the quotations] sometimes gives you ideas that don't show up in the forums you're searching.

Also, a tip. As obvious as it sounds, make sure you're flexible with your flight dates. There can sometimes be $100 difference between the two days. Also play with one way fares as these can sometimes be cheaper - especially when talking about international flights.

Using various dates and airlines, the cheapest I can find for you is $347.90 return.
In regards to knowing if your flight costs will come down, who knows. A party of 100 that are booked to fly on that day may all cancel. Unlikely, but it does happen. No-one can guarentee cheap tickets, not even the airlines.