The typical way to get to Sentosa from anywhere in Singapore to Sentosa is to take the MRT to Harbourfront and then take the Sentosa Express LRT from there. I suggest you use the website gothere.sg where you can just enter something like "City Hall to Sentosa" and it will then show you different options to get there, including the prices.
There are of course other options for getting to Sentosa, including taxi, bus, cable car or walking. Note that even if you walk you'll have to pay S$1 entry fee (compared to S$4 with the Sentosa Express). More information on getting to Sentosa can be found at their website. UPDATE: as of 2018 the walk-in fee has been waived until the end of the year. This has happened for the past couple years, and keeps on getting extended. So for now it's free to walk there.
The arguably most convenient and cheap way of getting around Singapore is to buy a EZ-Link card. Those are accessible to foreigners, transferable and widely available. If coming from the airport, you should be able to buy one at one of the convenience stores at the airport or at the train station there. You can then top them up at any MRT station. Note that tickets are cheaper when you use a EZ-Link card as compared to a standard ticket. As far as I know, you can also use this card to purchase your ticket to Sentosa for the LRT.
There's other options, such as the tourist pass, which allows for unlimited travel. However, in my experience you need to ride around a lot for those to be worth their money. As an estimate, consider that the maximum amount you can be charged for a trip island-wide is somewhere around S$ 2.5, most shorter trips will be cheaper.