If you're doing a lot of sightseeing on the island, car / motorbike rental is the way to go.
Almost every small - mid range hotel / bungalow operation will have motorbikes for rent, and some will have cars.
There are also plenty of Internet cafes / travel agents on Koh Samui that provide the car and motorcycle rentals. They require a deposit and proof of driving license. Some will keep your passport as well. If you're hiring a car make sure you look for damage before you set off and get a written statement that this was done before you hired the vehicle.
For a group outing round the island, get a 4 wheel drive to negotiate the off road tracks. If you just want to zip about, then hire a motorbike. Discounts are always available for monthly rentals.
Make sure you keep your drivers license with you, and if you are riding a motorbike wear the helmet provided. There are frequent police checks and you can be fined if you're not wearing one.
Chaweng, Lamai and Na Thon have parking restrictions because of the width of the roads. On odd days of the week you must park on one side of the road, on even days you must park on the other. Have a look at the road signs and they'll tell you which side to park.
Driving on Samui is not as hectic as Bangkok, but its no walk in the park! The roads are good, which is a blessing -
there are plenty of other obstacles to watch out for. Motorbikes, songthaews, taxis, bicycles, dogs, cats and pedestrians all jostle for space on the road.
Keep a sharp eye out for anything / anyone cutting across your path which happens frequently. Accidents usually happen on the main ring road where it's tempting to go fast, particularly outside Tesco, or at the crossroads that leads to Chaweng Beach road where the traffic bottlenecks.
Please be careful in these places.