I like the train idea but you would have to figure out in advance a place to store your luggage. I like to travel light when camping. I think that bicyclists are allowed in the park but private vehicles are generally not. The bus system seems to be the only realistic option to get transportation to trailheads or whatever.
The big trend now days is to rent small RVs and travel the state with them - I think that is a great idea if your budget allows. This style allows you to cover a lot of territory without worrying where to store your gear. One big variable is how much time you plan to spend on this trip. Another is how accustomed you are to camping in remote areas on your own.
Depending upon how much time and money is at your disposal you could make a great road trip out of it.
A suggested route would be to arrive in Anchorage and pick up your RV. head south direction Seward on the Kenai Peninsula. Follow the Turnagain Arm down to Girdwood watching for Beluga Whales on the way . Fuel up in Girdwood and drive down to Seward - check out the Reurection Bay and the Exit Glacier.
Then Head to Kenai / Soldotna where you can park overnight at Meyers or Walmart parking lot. Check out the Kenai river and great tour center in Kenai and on the river in Soldotna.
When you feel ready head south direction Homer but for sure stop at Ninilchik - an old Russian Settelment on the Cook Inlet. I caught a halibut off the beach there about 40 years ago ha ha - true (lure - Pixie pink stripe). Continue onwards to Homer (great little seaport on the Resurrection Bay - get a campsite on the beach. Build a campfire from driftwood on the beach if you like stay out all night if the spirit moves you (no one will care). be sure to check out the Islands and Ocean Visitors center.
Head back to Anc and resupply at Wasilla (food and groceries)go to Talkeetna as mentioned above. If budget allows hire a flight to Denali - especially if the weather is clear (you will not regret it). or drive up to Denali park and do the bus thing for a few days. The Denali thing would depend a lot on how much experience you have in wilderness camping. the bus thing is adequate for most.
If you feel like a trip to Fairbanks it might be ok but many would not be able to grasp the vibe there. Good pipeline info park and the Chena Hot springs 60 miles northeast.
note: a great place to make contacts is at Sven's Base Camp there off of Davis road in Fairbanks - I have met incredible travelers there including a Swiss guy that took a year or more to pedal his bike across Siberia, it is not unusual to meet motorcyclists that have just arrived from Patagonia on a round the world trip- this is what travel is about to me - meeting people like this
taking the long way back go to Delta Junction - turn direction Glennallen - if you want head south direction Valdez. - take the Edgerton cutoff to Chitna and drive about 60 miles to McCarthy . parking this side of the river - then cross over and go to Kennicott - pretty cool. Glacier and old copper mining town. I was walking up to the glacier a few years ago and was watching a beautiful cinnamon colored black bear when a shuttle of tourists roared past never glancing my way - weird - such a beautiful bear but everyone was to busy to look.
Anyway Thompson pass on the way to Valdez is memorable also (huge snowfall in the winter) Valdez is the terminal of the Alaska Pipeline - beautiful area. You and your RV could take the ferry to Whittier Alaska on the Kenai Peninsula if you felt like it. (it travels about 35 to 40 Knots per hr) passes what is left of the Columbia Glacier. Or you could drive back up to Glennallen and drive west on the Glen HWY to Anchorage from there (passing the Matanuska Glacier which is a nice stop over as well-largest Alaska glacier reachable by road)
return your RV and jump on the plane !
pardon my rambling - ideas only