Llŷn - spelled 'Lleyn' in the English language, a peninsula whose name is derived from the same Celtic root as the Irish province of Leinster
It would probably be best to learn some basic Welsh as you might encounter people who only speak Welsh. Although most local people are bilingual and can speak Welsh and English including all tourist shops and popular restaurants.
Plas yn Rhiw - beautiful, though not large, National Trust property east of Aberdaron
Penrhyn Castle National Trust - a mock-Norman edifice of the 19th century. Can be considered well over the top in the desire to impress but the kitchen quarters and the walled garden are particularly to be recommended. 
The Spinnies Tow splendidly situated bird hides administered by the North Wales Wildlife Trust. Birds frequently seen include little egrets and kingfishers. Near the coast immediately east of Penrhyn Castle estate.
Parc Glynllifon Lord Newbrough's estate south of Caernarfon - the arboretum and a modern amphitheatre are particularly worth seeing.
Ouzo and Olive, 14 hole in the wall street caernarfon (by the castle in Caernarfon). 6.30pm-9.00pm. This is a very nice Greek restaurant that is great for tourists. Very peacefull and relaxing. Open all year round.
Pale Hall, Palé Estate, Llandderfel, nr Bala, Gwynedd, Wales, LL23 7PS. UK, ☎ 01678 530285, . Palé Hall is a privately owned luxurious Victorian Country House Hotel near Bala in North Wales, set amongst 16 acres of gardens on the edge of Snowdonia National Park. Palé is a delightful discovery for discerning visitors - a lovingly restored Victorian country manor set in refreshingly tranquil and beautiful surroundings.