You can't see the borders by train - I think there are coach tours, but really you want a hire car. There are a number of amazing castles or upland walks but they aren't near anything else so there will be a fair amount of travel between sites.

Lake district is very nice - get down to Windermere (although if the weather is crap you may wish to rethink this bit - you just won't see anything if the cloud is at ground level)

If you are heading to Arran, you may wish to visit Culzean Castle before you go across (it is in Ayrshire and looks out towards Arran) and the Robbie Burns museum, and then get a boat across to Arran from Troon or Irvine. Brodick Castle on Arran is worth a visit, as are the standing stones at Machrie Moor.

I'm not sure of connectivity between Arran and Islay, but you do need to get there to enjoy some of the finest peaty whiskies in Scotland.

Oban to Skye is a great drive - absolutely beautiful.

Sounds like a great plan - just remember the train in Scotland is fine between major centres, and okay if you aim to get to a town on one of the main lines, but away from that things get a bit sparse.