I would imagine a ferry to the mainland would be the first step, but I need to know which part of the mainland to head to and how to travel from there. Are there buses or trains heading that way??? And how long will it take?
From what I know there is a regular service offered by at least 2 airlines direct from Samui to Phuket. If not available or out of your price range there are apparently a variety of options to get you across the mainland and most tour operators in both Phuket & Koh Samui could fix you up very easily with a ferry and bus. about 10 years ago
I think the two mainland ports that operate ferries to Koh Samui are Chumporn (I've been there) and Don Sak. Don Sak is close to Phuket, and from here it's a 6 hour bus ride to Phucket. You should be able to get from Koh Samui to Phuket in one day.
There are many travel agents in Koh Samui (esp. in the main beach at Chewang) who could book you the ferry and bus to Phucket. They take a small commission (perhaps 200 baht) but can save you the hassle of trying to book yourself at the Pier and Bus stations, and will also organise pickup from your hotel.
I would recommend that you book at least a day in advance, as certain days seem to be popular with crowds going to and from these two areas.
Also beware of paying extra for the 'VIP' bus which is apparently more comfortable than the A/C bus - in many cases there is no VIP bus and you end up on an A/C bus anyway. about 10 years ago