So far it is looking like a higher snowpack year. For what it is worth, NWS long term predictions is for more than usual snow. Western Wyoming is already getting quite a bit of snow in the mountains.

I lived in Wyoming many years. Wyoming is very windy; actually, most of Wyoming is always windy! Laramie area is particularly windy. Be sure your tent is wind-worthy up to 50-60 mph winds. Wind-proof layers are handy. Be sure you stove has a wind screen.

June snow tends to have a crust that you can hike on top of in the morning, but it collapses by noon. Plan most of your travel for the morning. After about 2PM you will likely be post-holing. Light hikers or boots will keep feet warmer than trail runners. Knee high gaiters are essential.

Gros Ventre in late June will still likely have snow. Good thing is that even if you have to walk on snow, you can usually find a dry spot to camp. Nevertheless, a sleeping pad with an R-4 or more rating is good, in case you do have to camp on snow.

For river crossings, be sure you have trekking poles and sturdy wading shoes. You can just wade in your hiking shoes. I find that taking an extra pair of insoles helpful in this case, as well as a minimum of 3 pair of socks. Wade in shoes without insoles and socks. Dry one pair of insoles while you wear the other. Shoes will get soaked if walking much in snow, so you simply have to deal with wet shoes. Be sure to have one pair of wool socks dedicated for sleeping.

June weather can be quite rainy; or wonderful. Good news is that mosquitoes usually do not show up above 8000 feet until July. You can also have daily snow squalls in June at 8000 feet or above.

I would skip the rope and heavy climbing gear and instead be sure to take sufficient clothing. You will get wet and may not be able to dry out clothing for days. Take sufficient fuel so that you can always have hot drinks. Waterproof matches also good. Skills such as knowing how to build a fire with wet wood is also useful.

Trails will likely be covered with snow sometimes. If not an expert navigator, be sure to carry a GPS.

You will pack differently for the lower elevation trips and the higher elevation trips. Do not try to have a one-size-fits-all set of gear.

If you are truly a beginner, skip any thought of "climbing". If conditions become such, simply turn around and go back.