An interesting trip close to Anchorage is the Hatcher Pass -Independence Mine area
. It's a quiet, somewhat isolated drive that has lots of varied things to see - and thoroughly Alaskan. I'm sorry I don't have a shot of it, but next time I go, this will be my first stop.
Denali is great choice if the snow has thawed enough. May is break-up time and early May will be quite different from late may. If you can get to Denali, you should. It is absolutely unique and you will never forget it. Take the bus all the back to the end of the road at Wonder Lake (it will take all day) and come back the next day. Wonder Lake is the classic Ansel Adams view of Denali (the mountain itself). You can get on and off the bus any time you like and catch another at a different time. The shot I'm putting up is from the Polychrome Pass area.
The Seward Highway down the Kenai Peninsula
has many very interesting places. The whole road is about 200 miles long and ends at the town of Homer. The Chugach State Park and National Forest are both very nice. The town of Seward is the starting place for Kenai Fjords National Park, The Holman Glacier, Resurrection Bay, and many wale watching tours. Kenai is an interesting town as is Soldotna and, if you like to fish, this is the place. Along the way there are many interesting stops and the road is very good. Even the drive is beautiful. I would take 2-3 days but many of the places are close enough to Anchorage to make several day trips rather than one longer trip. The photo I've attached is from the Crow Creek Mining Camp near Girdwood - about 40 miles from Anchorage.
The area around Anchorage will be rainy and cloudy a good deal of the time. But, if you move toward the interior (Talkeetna, Denali), there will be less rain. The Kenai Peninsula will also be very wet. You will love Alaska. I grew up in Fairbanks. My first darkroom, in the basement of our house there, was washed away in a flood in Fairbanks in 1967. I've been away since 1972, but you never get Alaska out of your heart. My last visit, my wife wanted to move - until I reminded her they have winter. We'll be returning again soon.
If you would like to see more of my photographs of Alaska, here's a link:
http://myrsphoto.500px.com/alaska
Enjoy your trip. I am envious!