I find regular hikes up Camelback and Piestewa Peaks in Phoenix are fantastic for good overall fitness. The west-side approach up Camelback (via Echo Canyon) works out some of the lesser muscle groups a little more since you'll be stepping every-which-way. Hike these with a full pack and maybe a couple times in a row.
For alpine, then yes, you may need to locate a guide and try a couple climbs to learn how to handle yourself on snow slopes and glacier. Unfortunately, AZ's biggest peaks don't emulate what you may find in Europe very well. For that, probably the Sierra Nevada in California is better. Still, hiking up and down Humphreys Peak near Flagstaff (el 12,633 feet) will help you learn how you handle higher elevations.
Some of the European country highpoints are low hills, so walk out into a field and look bewildered. That will be what you should expect over there. Make sure you get all those little tiny countries like Andorra, Liechtenstein, San Marino, Monaco and the Vatican City, too.