I can't speak to international climbing destinations in April-May.
For international non-climbing trips in April-May I recommend Turkey, without reservation.
May is the best month for hiking in Scotland. We got pretty sick and tired of the grey weather and rain, but many people love it there. The Cape Wrath Trail is worth assessing.
And, of course, ANY trip in southern Utah is fabulous in April-May. You live in Utah, so you know this already. Since you've got climbing skills, you'd be able to do any of the routes in Steve Allen's "Canyoneering 2: Technical Loop Hikes in Southern Utah", or the "Overland Route" described on page 306 of Steve Allen's "Canyoneering 3: Loop Hikes in Utah's Escalante". These routes are not suitable for people who lack advanced navigation skills and class 5.0 climbing skills. We have photos posted for only one Southern Utah trip -- http://amyl.smugmug.com/Backpacking/UT-OR/200904-Escalante-River -- but we've taken ~10 trips down there and we can offer ideas if you aren't already very familiar with the options.
April-May is too early for the Pyrenees. However, we've thoroughly enjoyed some of our trail hikes in the UK and France. From your comment about climbing and mountaineering I'm guessing you might not be interested in these. If you're willing to try a trail hike, then the Southwest Coast Path would be my first choice in April-May. Staying in B&Bs for a month would be outside your budget, but if you're willing to stealth camp it would not be a problem. And stealth camping in England is often as simple as asking the bartender at the village pub where you could set up your tent. http://www.southwestcoastpath.org.uk/
We've chosen the Alpe Adria Trail for our May 2014 trip. It is a valley and low pass route into the Alps that is suitable for a spring trip. If the snowpack is low you could do some high country side trips from that trail. Just a thought.