the best way to get around in Hokkaido is Public transportation, including train, bus and taxi.
I certainly don't recommend it for the entire stay, but for some parts of the trip, where public transportation is inconvenient, it might be a good option.
Inside Sapporo, however, it is probably better to use taxis instead of buses and subways, because walking and some stairs is often involved when using the subways and buses.
Otaru is relatively easily accessed by train, but for medium and longer distances within Otaru, a taxi might again be the best way of getting around.
As for Asahidake Onsen, note that besides hiking there is not much to do around the mountain resort. Nor is it a convenient base to explore the surrounding sights. I think that one night is enough. Two nights at the most. Access by train and bus via Asahikawa is quite convenient, but look up the timetable in advance, as bus service to Asahidake Onsen is infrequent.