What is the best way to see Alaska? Plane, train, auto, ship, dog sled?

I don't know if I can say what the "best" way is because I have only been there once and done it one way.  However, I LOVED the way we did it.  We flew in and out of Anchorage and drove.  We started with 3 nights in Healy which allowed us 2 full days to explore Denali National Park.  We rode the park bus the first day and did not get off, the second day we got off the bus and did some hiking.  We then drove to the Girdwood area for 2 nights.  There is some great scenery and hiking in this area.  Recommend the hike to the hand tram.  Next up was the Homer area which was amazingly beautiful.  We did a tour by the Alaska Coastal Study group and had a blast.  Our final destination was Seward, also for 2 nights.  Here we did a small boat tour with Saltwater Lodge that was 9 hours into the Kenai Fjords that was an experience of a lifetime.  I could talk for hours about or trip to Alaska it was just breathtaking.  One thing I would have liked to do was take a bear viewing trip by plane.  My husband was not having the tiny planes though so I had to pass.  

Avoid the large cruise ships.  Fly to Anchorage, take the train (upper level) to Seward and enjoy a day cruise there, as well as a visit to see the iditerod training center (Pet a puppy).  Take train to Talkeetna and fly over (and land on) glaciers.  Eat yummy food.  Take train on to Denali National Park and from there, take the bus into the park.  Avoid the large hotels where the cruise ship people swarm.

All of these folks definitely seem to understand how big Alaska is. Reality is, the best way to see Alaska is by all of the above. You'll never see all of it. I've lived here almost 30 years and have done my best to visit different parts of my state and still have barely scratched the surface. That being said -- if you don't cruise the Inside Passage or take the Alaska Marine Highway (state ferry) then fly into Anchorage, there are great restaurants and several great museums. Now, if limited on time... choose to go north or south. If you go south, stop in Girdwood at the Hotel Alyeska and take the tram up the mountainside for some amazing views and a little bit of mountain top hiking. Next, drive south to Seward, hit up Exit Glacier for your glacial hiking experience, then head south to Soldotna (Alaska) for a little bit of fishing on the Kenai River, then maybe Ninilchik for a visit to a Russian old-believer Orthodox church and some great halibut charter fishing. If you're digging the Old-believer vibe, visit Nikolaevsk for more Russia-in-Alaska history. Then keep going south just a little bit further and visit Homer. Homer has a ton of great little ed and breakfasts or VRBOs. From Homer you can visit Kachemak Bay State Park and a bunch of hiking trails, or just stay on your water taxi and see the Bay. I'd recommend Mako's Water Taxi, they've been around forever. The restaurants are diverse, often organic and locally sourced, as the seafood would be also... And if you have any time left - do not miss booking on Danny J Tours to go over to Halibut Cove to walk the boardwalks in the awesome little artsy community and enjoy some good grub at the Saltry Restaurant. 

Good luck!

All of the above but since you have time constraints there are a few things you can do to get the most out of your travels.  You did not say whether you have chosen a jump-off or termination point yet.  We went one week on ship and two weeks on land via train and car.  The Alaska Railroad goes from Seward in the south to Fairbanks in the north but only operates until mid-September.  If you use it, buy the Goldstar service.  You have choices of many drop off points including Denali National Park. If you were to stay near Denali I would suggest the Denali Dome Home Bed and Breakfast.  The only place near Denali National Park open year 'round.  Others only open during the season so that alone should tell you to whom they cater.  Research it and read the reviews.  Great hosts and locals who really know the area.  Stay away from "Glitter Gulch", as the locas call it, for your accomodations.  Visit Jeff King's (four time winner of the Iditerod) Husky Homestead.

If you don't use the train it is very easy to drive but you do miss out on viewing the passing scenery at a liesurely pace such as on the train.  Plus, the train has locals for guides to point out places of interest.  And the roads do not always take you via the most senic or fastest routes.  Remember, planes, trains and boats are the common modes of transportation.  There are more licensed pilots in Alaska than there are drivers.  You can't even get to Juneau (the state capital) by car.  

Look at Alaska Heritage Tours for a Kenai Fjords (Seward area) cruise.  Well worth the money.  Dog sledding at that time would probably have to be on a glacier because it won't be cold enough yet in most areas.  Check out the Knik River Lodge in Palmer, about 30 minutes from Anchorage.  They have helicopters that fly out to a glacier.  Their cabins are small and their restaurant is a Mongolian Yurt.  Food is good.  If you get to Skagway take the White Pass Railroad & KIondike Highway train.  They will stop and let you off to hike and pick you up on a return trip.

In Anchorage, you should go to Moose's Tooth Pub & Pizzeria.  Wonderful pizza and great beer, brewed on the premises.  Also in Anchorage visit the Alaska Heritage Museum (I think that is the name but the hotels will know for sure).  Very fascinating and a great historical picture of the state.

The only sure thing is that you will see some of the most majestic scenery in the world and come away with a new perspective and respect for the sheer vastness of the state and the friendliness of the residents.  Nothing like it.

Consider a naturalist-led expedition cruise.  It's like being on a safari.  Usually less than 40 passengers.  Kayak, zodiac, hike, get really close to the wildlife, glaciers and native Alaskans.

Last summer we did a car trip throughout southeast and I agree with the other answers. If you can take an eco tour, do it! We added a crazy overnight ferry ride from Homer to Kodiak  which turned out to be our favorite place. If you make it to Seward  the Sea Life Center is wonderful and you can arrange for a puffin encounter (several other choices but really who could resist a puffin?). You are going to LOVE Alaska!

I would say Alaska is spread out and by boat you would see the most things and still have adventures when you get to ports.

www.powercouplelife.com