well, do you like to hike/bike/and climb, or do you like diving more? If you want to go on some awesome hikes, staying near Anchorage would not be a bad idea at all. The Chugach State Park has got some of the best hikes in the state in terms of beauty, range of difficulty and length, and convienience of access (great trails/great trailheads). There are plenty of trails for either walking or biking depending on what you are into.
If you want to dive, get out of Anchorage pronto. You can reach Whittier, Seward, or Homer by car. Juneau, Sitka, and Ketchikan are also pretty "popular" dive spots-but if you start in Anchorage, you'll either have to fly or boat into these places.
Something to consider about the timing of your trip, in May, Denali is likely to still be pretty snowy. In late May, around the Chugach State Park (near Anchorage), many of the trails are just starting to become totally snow-free.
A good trip idea that I give to alot of people is the Serenity Falls Hut bike out at Eklutna Lake. This is about 45 minutes out of ANchorage, and you'll need to rent bikes to do it unless you want to spend alot more time walking.
The trip startes at the end of Eklutna Lake and follows the bank for about 6 miles, then heads off into an absolutely magnificently mountainous valley. You'll bike another 7 or 8 miles through some beautiful Alaskan forest with breath-taking views ot the mountains the whole way until you get to a pretty large cabin near the back of the valley. The cabin has a wood-stove and gas-lanterns (usually it is stocked with fuel), along with bunks for sleeping. There is no electricity, phones, or cellular reception of course. The amazing thing about the cabin is that one whole wall is entirely made of windows giving you a truly spectacular view of several extremely rugged mountains and part of the Eklutna Glacier. You can expect to see some pretty amazing wildlife along the way, including sheep, goats (yes there is a difference), bears, moose, and smaller creatures like porcupines and such.
The cabin costs something like 40$ to reserve a bunk, and the proceeds go to fund the state park cabins, trails, and other neccessary elements that keep the area enjoyable. You can reserve the entire cabin for something like 120 bucks or so, but beware! The cabin is a major hub on a fairly popular, often week-long ski-trek called the Eklutna Traverse, so there are sometimes a few folks bunked up there either on their way to the glacier or on their way back from it....and some of them won't give a damn whether you reserved the whole cabin or not....sometimes people NEED the cabin on their way down from the glacier. I've never booked the whole cabin, but a few times have still been the only one there anyways. More often than not it seems like if you are the only reservation on a given night, you are the only ones at the cabin-so my advise would be to save your $$ and only reserve bunks. It'd be a bummer to pay for the whole thing and then have to fight off a bunch of trip-weary mountaineers who don't want to give up the only shelter for miles just so you can have the big place to yourselves.
Anyhow, you can check dates of availability by going here:
Public Use Cabins, Anchorage Area and calling the number listed under "Serenity Falls Hut." They used to have the availability schedule online, but it looks like they don't anymore.
Well, I've spent way too long on this, so good luck.
btw-here's a pictoral report of a Serenity Falls trip I did a couple years back:
http://chugachpics.tripod.com/serenity/serenity.html