Most people will arrive here by car - for which there is ample car parking space at the entrance to the village. Its a small enough island to walk around but for the less mobile there is a minibus shuttle service from the car park, through the village to the castle. Details on the web link below - times will vary with the tide tides and castle opening times - plus no service on Fridays!
Driving up north to Lindisfarne is the best way 'cause you get to see a lot on your way :-)
You drive north of England along the main A1 roadway to the crossroad at Beal - that's where the causeway is about 5 miles east of Beal. And Beal is half-way between Newcastle and Edinburgh, that's about 8 miles south of Berwick-upon-Tweed.
Berwick is also the place for you to get to if you're travelling by train.
For area map and other transportation info check Lindisfarne Travel Information.