Trains and buses go lots of places in the British Isles but you're limited to their schedules. We used bicycles + buses in Western Ireland in 2011, but the cycling itself was a big part of the trip. Planning our Scotland trip for this summer, we decided we needed to rent a car for maximum flexibility on routes and timing. As Rick notes in his book, for Skye you need a car, and much of the Highlands & Islands appear to have infrequent bus/ferry service.
We only have 2 weeks this time, and even with 3+ weeks, I don't know if you're going to be able to see "all" the sights in any one of those countries, let alone all three! Within cities and going between major cities, public transportation should work for you. If you're going to more remote places and sights, you may need to consider a day tour or other options. Happy planning & travels!