I tend to agree with the above.

Hokkaido can absolutely be seen quite nicely by train, with a few buses thrown in for the more remote locations like Asahidake and Shiretoko and for local transit within specific cities.

I looped the whole island, mostly by JR train, in under 10 days. That included Hakodate, Sapporo, Asahikawa and the national park region above near Asahidake, Abashiri, the Shiretoko Peninsula, the national park/lakes area above Obihiro, Kushiro and a lot of beautiful scenery between those points. To make it even more of a rail trek, I started and ended at NRT, not CTS, stopping in Sendai on the way north and Morioka on the way back south. A fair number of meals were eki-bens on the trains. :-D

For doing that sort of thing, a Japan Rail Pass is very helpful. I also got a lot of help from the book "Japan by Rail," which is available from Amazon. Hokkaido is getting much more popular, so I would strongly recommend making train seat reservations as soon as you arrive in Japan and know what routes you want. I had made up a list of all trains I needed before leaving home. HyperDia is very helpful for that. The little local trains are a joy and rarely full but the shinkansen and limited express trains were often full, even several years ago when I visited Hokkaido the first time.

https://www.amazon.com/Japan-Rail-3r.../dp/1905864396

http://www.hyperdia.com/en/