You might as well just catch a flight to your first destination. It will be cheaper than renting a car and driving (which could take days, so you would also be paying for hotel rooms). You are going to have to fly to Alaska anyway.
If you go to Alaska, I would recommend Anchorage. There is more to do there, it is usually cheaper to fly there, and there is more within a couple hours' drive from there than Fairbanks. Also, go in the summer. While it is more expensive, there is a lot more tourist-oriented stuff open in the summer than in the winter. Unless, of course, you want to do winter stuff. If you want to see the northern lights in winter, then I would recommend Fairbanks over Anchorage, and head out to Chena Hot Springs.
I'm not sure what your interest in Maroon Peak is. I would say head to Denver or Colorado Springs and use that as a base to hit things in the area.
Keep in mind that trains and busses in the US are a lot different than what you find in Europe. Intercity busses are generally only used by people on the lower end of the economic scale. We have one passenger train company, Amtrak, and its service is relatively limited. However, if in Alaska, and you have the time, you might want to use the Alaska RR to go between Fairbanks and Anchorage. If you can handle driving on the right side of the road and can afford it, you might want to just rent a car and drive, or fly between cities.
UPDATE: Since driving is out of the question, I would say your best options are flying between cities or possibly Amtrak. Trying to go cross country by bus would be miserable. However, not having a car is going to limit your options at a few locations. You would have to research tours that will get you around each place. You can probably get around New Orleans pretty easily, and there are plenty of tours in Alaska to get you around.