The Wrangell-St. Elias National Park is one of the most primitive parks in the world with some stunning scenery that can only be reached by bush planes. Its' chief attraction is also its' detriment to many people who can't afford the cost of flying to many of these destinations.
Thankfully, there is Nabesna Road, one of only 2 roads that go into the park. The road itself is rough and potholed but it is worth the time to do the entire road for sightseeing.
The Skookum Volcano Trail is a great choice for a relatively easy day hike with a great payoff. Parts of the trail are steep, but you get fantastic views once above the treeline.
The trail follows an alluvial fan of black spruce and cottonwood before emerging into a creek bed. After about 1 mile, the creek splits - follow the trail to the left which has rock cairns marking the way. The creek level changes throughout the year, but you should expect to do some hopping over rocks, and you may even get your feet wet, so plan accordingly.
The trail ends at the top of a pass, rewarding you with up close views of the volcano, and plenty of interesting geologic features. From there, you can either go back the way you came, or it's possible to make a loop through the pass. I have not tried that way myself, and my understanding is that trail is much more primitive.
If you're looking for a real primitive and pristine Alaska experience, and you're on a limited budget, this is a great bet.
This area is for experienced hikers. A working knowledge of topo maps and navigation is important. In this park, you cannot expect that you will be rescued or found right away. You should hike with at least one other person. A wise person will leave their itinerary with the ranger station at the beginning of the road. I urge you to go to my Education page and make sure you have all the necessary skills covered.
This is definitely mosquito country, so bring the DEET!
Be aware that several creeks cross under the road. In heavy rain, they may overflow possibly trapping you in the park until they recede. Pack some extra supplies in your vehicle just in case you have to stay over for a few days.
If you have some extra time, this would be a great opportunity to bike the entire 40 mile road. It's rough, so use a mountain bike.
There are some other great hikes along Nabesna Road that I would love to come back and try some day. Rambler Mine was short and scenic, with some great historical value. I've also heard good things about the Trail Creek/Lost Creek loop, and the Caribou Creek Trail.
For more information on this area, I heartily recommend Hiking in Wrangell-St. Elias National Park by Danny Kost.
There are several single campsites with their own lakes and outhouses. Get there early so you can claim one for your own.
The drive from Anchorage is a great road trip. The Glenn Highway is one of the most scenic in America. At mile 100, I recommend making the Matanuska Glacier Rest Area your lunch stop. Since the driving time is under 5 hours, you may also consider taking a side trip to Matanuska Glacier where you can actually walk on top of it.