Stay well clear of Lands End - it has been ruined, well and truly.
West Penwith is beautiful, the road from Sennen to St Ives stunning if somewhat twisty, see Sennen, St Just, Zenoor (lovely pub in the Tinners), great walks along the cliffs. There's also St Bunyan inland.....
South coast, Lamorna Cove, the Minack Theatre, Mousehole (lovely bistro in No 2 Fore Street), Newlyn - traditional Cornish icecream at Jelberts, right on the bridge there, Helen Glover's family own it, she of 1st gold medal fame at the Olympics. Penzance, the main gallery for their Stanhope Forbes collection.
There are a lot of good galleries around, Kurt Jackson in St Just, the Leach centre as you come out of St Ives towards Zennor, smaller galleries at Halestown (between Penzance and St Ives).
Take your paints if you are artistic, the countryside is savage but stunning and a major input into both the St Ives and Newlyn Schools - lastly, St Ives does not accept cars into the centre during the summer - part at St Erth and take the coastal rail in. The Tate, St Ives School of Art (where I trained) and galleries along Back Street West, the Hepworth Gallery are worth visiting, and the town is much improved re shops and restaurants - the Porthminster Cafe and Seafood Restaurant on the main front are noteworthy. Also, for the arts, see if the Town Hall is open as they have a magnificent collection of Alfred Wallis paintings, a naive artist who was a fisherman, the church also beautiful....