Well, the Sockeye normally start running in July. There are a lot of fun, and in my opinion are the best eating but can be very picky about what they'll bite. August and September are the time for Coho fishing, the most popular of the sport salmon. They fight hard and will readily take a fly. From late July through late August there is also lots of Dogs and Pinks coming in. Both of them will take almost anything you throw at them, but the Pinks aren't much in the way of fighters. Dogs can be a ton of fun though. And I can't forget the legendary Chinook, they normally show up here in early July.
For a diy trip, from Anchorage you can reach plenty of places on the road with a few hours of driving. I don't fish up that way though so I can't provide much in the way of specifics for there though. I'm here in Cordova, pretty small town with some great fishing within short driving distance. There's also some people here who have fishing camps or offer day trips.
On the trout side of things, I understand there are tons of small lakes around Anchorage and in the valley with trout and pike. There are some cutthroat and dolly's around here, and maybe a few grayling left. We're mostly known for the salmon though, not the trout. I've never managed to hook any of size around here, but I've heard they are out there.
Here's a ADF&G page with tons of good info for you:
Sport Fish Division - Alaska Department of Fish and Game
And I guess I might as well throw in a plug for Cordova with a link to our Chamber of Commerce page. Might be helpful if you consider coming here.
cordovachamber.com - Home
Hope some of that was helpful for you!