Barcelona
A beach, a mountain and Gaudí, all reachable on foot from a single apartment. Stay 4–5 days.
Peak daylight. Iceland and the Nordics get near-endless light, the Rockies open up, and Europe is warm before it's unbearable.
Early June is much cheaper than late June across Europe — schools break up in the second half of the month.
Waterfalls, black sand and geothermal water, all within a day's drive of the airport. Stay 4–5 days for the south, 8–10 for the ring road.