As Mike has told you, there are no trains on Skye. Any rail journey from Edinburgh will involve changes, either in Inverness if travelling to Kyle of Lochalsh, or in Glasgow and Fort William if travelling to Mallaig.
Bus schedules on Skye link the main villages, but basically run to a timetable designed to get kids to and from school so there are often only two buses per day and if you are really lucky, one at some point in the middle of the day. They don't go anywhere near what I call the 'honey pot' sights on Skye, Neist Point, the Quiraing, the Fairy Pools.
I think you would find it more satisfying either to take a small group tour with someone like Rabbies, or to rent a car. Morrisons on Skye have great service (I've rented from them when my own car was off the road). They will arrange to meet you off the ferry at Armadale (from Mallaig) or at the railway station in Kyle.
Most guests that we get at our own B&B stay for 3 nights and leave saying they haven't seen everything they wanted to see. That could be because we tell them about less visited places that aren't in the guide books! If you are not interested in hiking (long or short walks) then you will probably be able to see the main sights with 3 nights.
Another thing to bear in mind is that you are likely to have difficulties finding good accommodation if you are coming this year. Most places book up months in advance. There is a facebook group called Skye Rooms that it would be well worth you looking at as it hooks up providers with spaces with visitors looking for specific dates.