What are useful tips and tricks to use when searching through the auction house?
I'm not going to repeat the good points @Nevir raised about filtering on search.
Repair all equipment before using AH If you are buying items for someone and they have damaged equipment, you cannot easily compare stats.
Rare amulets/rings These are snapped up immediately when they show up with a low buyout (like 2000) - if you have filters set properly and see one of these, consider just buying it asap instead of viewing stats.
Bid-only items Usually these should be filtered out, but periodically check the list without a buyout filter in place. What you're trying to do is find some bid-only stuff that is
and then place a low bid on it. This will almost never work if you are looking for highest value possible of popular attributes like gold find, magic find etc., and the AH doesn't allow you to sort by expiration time which is a severe disadvantage.
However this is worth trying as you can get lucky and find some decent gear quite cheaply. It is easier to do with more unpopular categories like follower specials or class-specific items. Also, it is easy to place multiple low bids on items ... worst case, you have bought 4 rares for 250 each but its more likely that they get bid up later and you're lucky to get even one.
Recommendations The gear recommendations you get on the Sell page by default, are targeted solely towards your class. This also means that you may miss out on a good item for your current class, just because the system thinks it is better suited for someone else (or even if it is a perfect fit for an alt).
Choose the correct character When you have multiple characters, make sure to choose the one you're buying things for. I have noticed that sometimes, trying to compare with what you already have won't work properly if you last logged in a different alt -e.g. Wizard 20 is the toon who wants a sash, but last time you played Monk 6; items are filtered in level range for wizard but the comparison screen shows Monk sash. If this happens, just login the monk and then logout.
Using Filters - If you set multiple filters and want to void one of them, just select None in that box even if its the first one (others will still get applied).
The auction house is inundated with a tremendous volume of items; you will spend far too much time paging through items, and most likely you'll miss some great deals.
Narrowing Down the Competition
Once you've identified what you want, you're ready to narrow down your results to something that you can quickly scan. (Good deals are often picked up within seconds of being posted)