Broadford Airport, on the Isle of Skye, is 221 miles from Newcastle Airport but there are currently no commercial flights operating into Broadford.
The road distance from Newcastle to Broadford is 318 miles, with an estimated driving time of 7 hours 10 minutes, excluding breaks. (Source: AA Route Planner).
Train connections are extremely limited. On weekdays you need to depart from Newcastle at 0741 in order to arrive in Kyle of Lochalsh at 1558. That's a journey of 8 hours 17 minutes, involving 3 changes (at Edinburgh, Stirling and Inverness). From Kyle of Lochalsh you'd need to take a bus, over the Skye Bridge, to get onto the island.
Alternatively you could leave Newcastle station at 0929 for an 8 hour 14 minute train journey to Mallaig (changing at Edinburgh and Glasgow Queen Street), arriving at 1743. From there you'd need to take a ferry across to the Isle of Skye, and then take a bus to your final destination.
Chris