Even though there's many ways of getting around the island, the best way to see lots of things is to just walk, you can walk along the beach for miles, walk out of town, you never know what you will find, small shops tucked away, resturants loads of things.
Walking around at night is also a load of fun and is very safe, not once did we ever feel threatened or scared, every one greeted you with a smile, so don't be scared to use your feet.
Like what I've said at the title above, Walking is the best option when in Patong Area as the Patong area is quite a small area and there is not that much volume of vehicles unlike in bangkok so that traffic is nonexistent here. Most of the Areas are just brisk walking distance from each other and you can walk at a slow pace as to fully savor the sights and sounds around the area. (and if you are an el cheapo, taxi rates and tuk tuk rates are higher here than in bangkok by more than 100%). there are many things you can do while walking like buying souvenirs, eat food, hang around beside the beach, get a real tattoo or a henna tattoo, have a beer at one of the roadside bars and more. just be sure to bring your sunglasses to shade your from glare in the afternoons. If you plan to go to Karon and Kata areas and vice versa, then you need to ride a public bus or a tuk tuk or taxi as these are far from Patong Beach, unless you are a marathoner or just plain el cheapo.
To me, I find that the best way to get to know a place is through walking around the place.
Frankly, the main patong area is not that big. You can walk and shop at the same time or do a foot massage in between the shopping.
If you do not want to walk, you can hop on to the tuk tuks (they are everywhere) and bargain for a good price.
I don't recall seeing much metered-taxis but there are a lot of "private" taxis which you can take but must also prepared to bargain as well.