First and foremost for availability you have to plan well ahead of time, like 355 days ahead of the date you want to travel. The popular routes will have all the Avios seats taken as soon as the booking window opens up. Less popular routes will have availability at shorter notice , and some routes will have availability added through the year but is not guaranteed. Bottom line is be flexible with dates and book well ahead of time.

I find good value traveling to the UK or Europe in business or first. Yes, the taxes/fees are outrageous but at the end of the day they usually work out to be similar to the price of a coach ticket and the Avios effectively buys an upgrade to business/first. Remember Avios tickets also provide some degree of flexibility because you can cancel for a fee of $35 for each ticket up to 24 hours prior to the flight.

The taxes/fees can be minimized by booking two one ways rather than a round trip. Taxes/fees ex USA are significantly higher than those ex UK/EU. A round trip commencing in the USA will incur the much higher taxes/fees on both the outward and the return flight, booking two one ways will have the lower taxes/fees from the EU/UK applied to the return. There is usually several hundreds of dollars saving doing that, although if you have to cancel, you will incur two cancellation fees of $35 instead of one because there are two tickets involved. The high UK APD does not apply if departing from outside the EU and transiting in London for less than 24 hours, that is another way to save another several hundred dollars. If you want to visit the UK, go there first and then fly on.

Availability on AA to Hawaii in coach is currently still pretty good so that is an option, and as others have said anywhere else is hit and miss.