Public transport system is very good in Hokkaido, you can reach almost all of the places of interests by train or bus, so going F&E is no problem at all.
In fact, it's advisable to go Hokkaido F&E instead of tour. All of my friends who went there with tour groups have one major complaint - they never have enough time at the place they visited, it's always "rush, rush, rush" with tour.
If you know your way around Tokyo's complex JR/subway system, then you'll certainly have no problem in Hokkaido which is way simpler and straight forward.
To save cost, just make use of special tickets or pass offered, for eg.:
-
JR Hakkaido Rail Pass (3/4/5 days) - if you plan to travel to several cities within that 3 to 5 days.
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Sapporo-Otaru Welcome Pass (1day)- JR and subway within Sapporo & Otaru.
- JR Furano-Biei Pass (4 days) - use of JR train on the Furano Line and Twinkle Bus (tour) for Biei.
Make use of
Hyperdia.com to plan your transport need, it has all the timing & cost details and it's in English.
To drive or not, depends largely on your itinerary and which city will you be spending most time in. If your itinerary is mostly in/near Sapporo, then it'd be better to just use public transports.
It also depends on when do you plan to go to Hokkaido? In general, it's not advisable for us Singaporean to drive there during the winters as most of us have little or no experience dealing with snowing conditions and icy/slippery roads.
Having said the above, it's actually quite fun to drive in Hokkaido during my last trip. The roads are not congested (outside city), road signs are clear, parking are free (in most places) and the car navigation system is easy to use. Most importantly, driving gives you freedom and flexibility in planning your itinerary and making full use of your time.
Let me know should you decide to drive, I can pass you some guides (PDF) that I obtained from my earlier trip.