perspectives from a New Englander who moved here 6 months ago...
Anyplace is great to hike, though my first 14er taught me that it's best to be 1) an established hiker and 2) acclimatized for at least 2 weeks, if you want to have fun hiking all the way to the top of a 14er. if not, you can still do it on some, it just won't be as much fun. As others have noted, peakbagging is absolutely not mandatory to have fun.
June has snow above 12,000 feet most anyplace. The farther south the less snow, though not always. Your tolerance for snow may determine where you end up going. Places east of the Divide may get less and melt out quicker.
If late enough in June (say, after the 15th): go see wildflowers!!!!! Anyplace between treeline and, say, 12,500 feet. Treeline is 11 to 11.5K most places.
Take thunderstorms seriously if you're planning to be up high. Rise EARLY and be down below treeline by 1 PM. EVERY day. Exceptions are unusual between late May and Labor Day. In New England, you can scoot down in 20 minutes or less if you need to, but not in CO.
And locally, if you go to "Gahden o' the Gawds" (nobody here understands when I say that!), go at or before sunrise. The light is waaay better!
Enjoy.