If you're wanting to fish for salmon along with flying and sightseeing, the last two weeks of July would be ideal. I would either rent an RV or do the rental car/B&B option - both work - and would consider the following itinerary:
-Drive south along Turnagain Arm and through Turnagain Pass and the Kenai Mountains. Yeah, it's a bit busy in July because everyone else is also chasing salmon, but it's incredibly beautiful. Drive to Homer. Take a water taxi across the bay (Ashore Water Taxi is my favorite) and spend a day hiking. The next day (or whenever the weather is good), take a float plane ride and check out the brown bears on the other side of Cook Inlet. Drive back north to Cooper Landing and take a charter/float trip for sockeye salmon and rainbow trout on the Kenai River. You can do this on your own as well, but as someone new to Alaskan fishing, a guide will help. Drive north to Whittier. Do a combo trip with Crazy Ray's for silver salmon, halibut, and ling cod. A side trip to Seward is well worth the time as well. If you enjoy hiking, the Kenai Mountains have several spectacular hikes. If you like beer, there are several excellent breweries along the way.
Reset for a night in Anchorage. Check out the aviation museum, Lake Hood, lots of breweries and great restaurants.
Head north to Talkeetna. Catch a flight around Denali. Drive north to Cantwell the next day, then turn right and follow the Denali Highway. Stay two nights at Alpine Creek Lodge. Go grayling fishing. They're small, but they inhabit the most beautiful places on earth and are eager to take a fly or spinner. Go the rest of the way east across the Denali, then head south. If time allows, consider driving to Valdez (more saltwater fishing if you're still eager), or stop and fish salmon on the Gulkana or Klutina Rivers, or head east towards McCarthy (a truly spectacular locale, but it's a long and bumpy road). Then take the Glenn Highway back west towards Anchorage. Maybe stay the night at Sheep Mountain Lodge if for no other reason than it's a ridiculously beautiful setting. Oh, and you can hike on the Matanuska Glacier close by. Also, blueberries should be getting ripe right around then, and that whole loop from the Denali Highway around to Sheep Mountain is prime berry picking country.
When passing through Palmer, make it a point to stop at Bleeding Heart Brewery (again, if you like beer). Find one of us from this site to give you a ride up to Knik Glacier and Lake George. It's close to home and is a spectacular backyard playground for backcountry pilots.
Enjoy!