DougS Loc: Central Arkansas

In order of "Do not miss" Alaska:
#1 Katmai NP!!! Spectacular Scenery! And the brown bears can come quite close! You need (MUST?) allow extra days for visiting, as the flights over from Homer get cancelled due to high winds, quite frequently! I allowed 3 days, (we got to fly over on the 2nd try). You can do your fishing if you can't fly due to high winds.
At Homer, there is a small park/wetland area right before the spit, where we saw moose, an eagle nest, and Sandhill cranes with a colt. Moose can be/are in town in yards...
#2 #3 #4 #5, #6: Kenai Fjords NP Seward (Major Marine)(Whales, orcas, etc); Glacier Bay, stay at the lodge (and get the boat included option)(hopefully on a clear day...); Misty Fjords NM ("Island Wings Air"), float plane,again on a clear day; Wrangell (Alaska Charter and adventures) has Stikine River (and Stikine Glacier), Anan Bear Observatory, LeConte Glacier. Tracy Arm Fjord (Adventure Bound), out of Juneau;
Normally, along the coast of Alaska, the least rainy period is mid May-June.
You may see (probably!) wildlife at all of the above. Example: Bald eagles, porcupine, moose, black bear, ravens, sea lions, seals, puffins, gulls, brown bear...
At Katmai NP in June, the bears will be still along the coast (not at the falls). Our visit was along the coast, and it was spectacular! EVERYthing else was 2nd place!
Wingpilot's suggestions are spot on.
All of the tour operators are pricey, due to such a short season to make a living.

Wkndr Loc: Sitka, Alaska

AR Farm Gal Loc: Piggott, AR (NE AR)

AR Farm Gal Loc: Piggott, AR (NE AR)

Great suggestions! Exactly the kind of info we're looking for. Thank you so much.

cahale Loc: San Angelo, TX

I assume Juneau as your starting point, since you mentioned southern coast and the inside passage. Mendenhall Glacier is worth the effort, which actually isn't much (effort). Further south, there are side roads to minor waterfalls and streams. If you have time, after you take the 6+ hours inside passage to Skagway, travel up into Yukon and across the southern part to the highway going to Muncho Lake, BC. Lots of streams, waterfalls, and mountains. I captured pictures of red foxes, lynx, buffalo, bears, 3 types of sheep, and reindeer (caribou). If you have sufficient time, it will be well worth it. There is a place on Muncho Lake (Double G Services) that I like. The accommodations are rudimentary but comfortable, and the food there is excellent. Photo of Whirlpool Canyon on the Laird River attached. Enjoy.

Whirlpool Canyon, upper British Columbia.

(Download)

Robertl594 Loc: Bloomfield Hills, Michigan and Nantucket

Adak is a pretty special place. Half way to Russia, farthest western point of the USA. Hard to get there. Only 2 flights per week from Anchorage, at least there used to be. I was there a few years ago on business. Incredible place. Here are some photos. Definitely not for everybody however. https://pbase.com/robertl594/adak

MtManMD Loc: Beaverton, Oregon

A salmon fishing trip on the Kenai River was really nice for us. Go with a guide on a boat. It was not that expensive for family of 4. We went in August and 3 of the 4 of us caught our limit of Silver's. My son landed an out of season 25 lb. King.

SteveR Loc: Michigan

Consider an Alaskan Railroad Tour, there are several. The Denali tour is supposed to be a good one.

https://www.alaskarailroad.com/

TheShoe Loc: Lacey, WA

The best bear watching I have done was during a week with a chartered boat out of Kodiak. We spent a week around the five islands of the Kodiak Archipelago that are home to the Kodiak subspecies, the largest land predator on Earth, each with its own personality.

DougS Loc: Central Arkansas

Since you asked about June, the salmon will probably not be 'running'. That being said, there is a King Salmon run in June. We were invited to go 'early run' King Salmon fishing with our B&B hosts out of Wrangell, on the 30th of May. We also saw some being harvested inland, south of Tok, about a week later.
We saw a 'barn door' halibut leaving the marina at Glacier Bay. Most (virtually all?) of the small communities have charter fishing up and down the coast. We saw 'weigh-ins' while visiting the docks at Valdez and Seward. The Halibut were in the 20-30lb range, with some Rockfish, and possibly Lingcod. These were during the first 2 weeks in June.
Don't sell the Marine Highway short, it is spectacular scenery all the way! I spent most of my time on the outside decks, fore and aft! The only one that is overnight is between Wrangell and Juneau (ignoring Washington to Ketchikan, which is 2 nights), and the sunset over Petersburg is spectacular! I asked to visit the bridge, via the purser, and got permission! It is quite interesting! Much better than flying over all that scenery. Check it out!
If you need to use the Marine Highway, keep in mind they are subject to changing their schedule, they are supposed to notify you if you have a reservation, of any changes. They didn't notify me, but I checked in advance and made arrangements to fly from Juneau to Glacier Bay, with no issues.
We saw our 1st black bear along the Stikine River, several along Tracy Arm, 3 more after disembarking at Auk Bay (turned the opposite way from Juneau, as we were early).
Along the Inside Passage, you will probably have the 'tourist' area's of the towns to yourselves, since there will not be any large cruise ships in port (usually 3 or 4 everyday, with multiple thousands each!).

Approx. # Picts I took at each local, and time spent there (I actually spent more days at each location, but the major areas, and not travel time, are what I am counting here)(since I drove, these were actually over about 20 days): Misty Fjords/Ketchikan 500+; Wrangell and/ferry to Juneau 900 (2days); Tracy Arm 1200+(approx. 6 hours); Glacier Bay 850+ (1 day); Haines 300+ (3 days); Skagway 250+ (1 day); Seward/Kenai Fjords 750+ (2 days); Katmai NP 800+ (approx 6 hours)(guide also tried to find the wolves, just got a fresh footprint)(at least a dozen brown bears); Kenai Peninsula 450+ (2 days).

AR Farm Gal Loc: Piggott, AR (NE AR)

Great information! Thank you.

AR Farm Gal Loc: Piggott, AR (NE AR)

Thank you so much for sharing your information. It is very helpful! We just need more than 10 days or so. We're trying to narrow our options down to something do-able.

AR Farm Gal Loc: Piggott, AR (NE AR)

That would be amazing. I'll check it out. I imagine it will be too pricey, but I'll take a look. Thank you.

AR Farm Gal Loc: Piggott, AR (NE AR)

Thank you!

AR Farm Gal Loc: Piggott, AR (NE AR)

Sounds like we need to just spend the summer there! (I wish). Thanks for your help.