Hokkaido is vast in size, so allow plenty of time to get around and don't try to do too much if your time is limited. Many Japanese maps including the generally excellent Japan Road Atlas show Hokkaido with a larger scale than the rest of the country, which may make distances appear deceptively small.
By train
The train network in Hokkaido is by Japanese standards limited, although it's more than adequate for travel between major cities. However, access to many of the more interesting sites, such as Hokkaido's many national parks, will require either relying on infrequent and expensive buses, renting your own car, or trying your luck at hitchhiking.
Some convenient express trains include the Hokuto and Super Hokuto between Sapporo and Hakodate 3.5 hours, ¥8,590 each way; the Super White Arrow between Sapporo and Asahikawa 1.5 hours, ¥4,680 each way; the Tokachi between Sapporo and Obihiro 3 hours, ¥7,920 each way; the Super Oozora between Sapporo and Kushiro 4 hours, ¥9,120 each way; and the Super Soya, Sarobetsu, and Rishiri between Sapporo and Wakkanai 5.5 hours, ¥10,170 each way.
JR offers a special Hokkaido Pass (http://www.jalpak.com/rai...), separate from the Japan Rail Pass, which allows the bearer to ride all JR trains in Hokkaido, as well as most JR buses.
By bus
A cheaper if slower and less comfortable option than the train is using buses, which also cover all the areas not accessible by train. Sleeper services radiate from Sapporo to most corners of the island. Note that local bus schedules can be very sparse, so check them carefully to avoid being stranded.
By road
Hitchhiking is a viable option in Hokkaido, and due to the limitations of the public transport network it's not unheard of to see Japanese with their thumb out a very rare sight in the rest of the country. The major caveats are that even private car traffic can be minimal on some roads, and for half the year the weather is colder than the rest of the country.
See also: Hitchhiking in Japan
By plane
Due to its vast size and numerous outlying islands, Hokkaido has a fairly well-developed commuter airline network. The main regional carriers are JAL subsidiary Hokkaido Air Commuter and ANA subsidiary Air Nippon now operating in its parent's livery. Many turboprop flights operate out of the tiny Okadama Airport in central Sapporo.
By bike
Hokkaido is a cycling paradise from April to September. There are many bike paths and most main roads have very wide sidewalks. Also there are many beautiful back roads to get you where you want to go. Information in English is very limited, the best way is to buy a good map and plan by yourself.
By car
By far the most convenient way of getting around sparsely populated Hokkaido is by renting a car. This is especially so when visiting some of the national parks or onsen resorts. However, visitors not used to driving in snow should be careful in the winter, and note that speed limits are reduced significantly only about 80km/h in winter when the expressways are covered in snow. As such, give yourself more time to cover the same distance in the winter than you would in the summer.