I echo all of this! For example, last year on the Highline in mid-August, we had one day of clear skies in the afternoon (our first day), followed by waking up to mostly overcast skies that turned into light but persistent drizzle for a good while, with one good appearance of the sun in mid-morning before it all cleared out early that evening. The next two days were the typical Uinta summer weather of clear skies in the morning and evenings with lots of ominous looking storms flaring up and looming around in the afternoons, one of them that hit us with a bit of hail. Our last day out looked as if it was going to stay clear for the day. The week before that, I had my family up there and we had nice weather for the afternoon, but as evening came, the skies quickly filled out with nice storm that shut us in until much later in the evening. Skies cleared out through the night giving way to another beautiful morning.
July-August can be among the most beautiful times to be up there because of all the greenery and wildflowers, but you'll be lucky to have 1-2 days in a week where storms don't flare up at some point in the day (typically in the afternoons) and there may be one day in a week where storms fire up much earlier and hang out for most of the day, while another day, it can be great all day before a big storm rolls in to conclude the day. That's just the way the monsoon pattern goes. Same goes for any high alpine mountain wilderness environment out here in the mountain west. June-July is also when mosquitos are usually at their worst and many measures must be taken to keep them at bay before they taper off in August.
As others said above, September and early October can be beautiful in their own way. The green meadows will have gone brown and lost their flowers and color, but that is made up by the reds and yellows of some of the trees and shrubbery. Trails are much drier, fish are hungry, crowds are thinned out, and of course skies stay clearer (though an occasional system can still erupt and even produce snow on higher ridges).
In short, if you want to experience them now, don't let daily storms stop you. Bring some rain gear, have a good shelter ready to set up, and do your hiking as weather allows. The average day will give you beautiful, clear mornings followed by some afternoon thundershowers, before giving way to a clear, calm night, but as I just showed, some days can vary from that typical pattern. You'll almost always have some time of the day where skies are mostly clear with some beautiful scenery to enjoy and hike through though. If you find yourself above tree-line when you see storms start brewing, it's time to get back down below tree-line ASAP. To minimize lightning risk, you don't want to get caught up on a peak, ridge, or pass when the storms build up.
