Try using awardhacker as well. It will give you an idea of starting miles/points for award tickets and on what airline programs. Once you have that info, you can start researching on the airline websites to see more about the routes, how much availability, etc.
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If you have Amex, Amex.point.me is great as well and actually shows real-time award availability, though the taxes/surcharges are often off.
What I tend to find, though, however, is that for economy flights, the taxes/surcharges can be so much on many airline programs, that I almost feel like I am wasting my points and I'd rather pay cash. If you find a good cash rate (I often find rates for under $500 RT in economy to Europe), then you can also opt to book it through the Chase UR portal. You give up some control that way because they are a
travel agent, but it's an option.
I'd search for a list of airlines that have low/no fuel surcharges to Europe and cross reference that with your award hacker findings to zero in on some programs/routes you'd want to research more.
That's a lot of info- feel free to ask more questions.