The routes below are, in my opinion, those places that you should visit when coming to visit this so unknown country. Most are easily accessible on public transport and, except in high season, you shouldn´t worry too much about crowds.
1) DONOSTIA-SAN SEBASTIÁN, one of the most beautiful cities in Europe (in the world?), views from Igeldo mountain, pintxos in the Old Town, Chillida-Leku open air museum (www.museochillidaleku.com), the picturesque fishing town of Hondarribia.
2) ZARAUTZ, GETARIA (wonderful promenade almost touching the sea linking both towns), grilled fish at any of the restaurants in Getaria, ZUMAIA and its beautiful church, LEKEITIO, its port, its island and the incredible retable inside its church.
4) SAINT JEAN DE LUZ, BIARRITZ, BAYONNE-BAIONA, in the French Basque Country (also Petit Bayonne, on the other side of the river), great chocolates and gateau basque, apart from picturesque villages.
5) BILBAO, Guggenheim Museum (this was obvious!), Old Town, Funicular train to Artxanda for spectacular views of the city.
6) HANGING BRIDGE (Puente Colgante) of Portugalete, a Unesco World Heritage Monument, unique in the world, over 110 years old and running 24 hours a day.
7) VITORIA, Old Quarter (jewish) and a visit to the spectacular restoration of its Old Cathedral (Ken Follets based its novel The Pillars of the Earth on this cathedral, so they made him a statue!)
8) ATXONDO valley, where silence can be heard, spectacular place with the mountains and sheep in the background. And several wonderful restaurants.
9) URDAIBAI Biosphere Reserve and the beaches of Laga, Laida, the town of Busturia, Mundaka (surfers´ paradise), the sea estuary, the caves of Santimamiñe and the PAINTED FOREST OF OMA. Also the steep fishing town of Elantxobe.
10) The walled towns of LAGUARDIA and LABASTIDA and the wine region of Rioja Alavesa. Not to miss the guided visits to the wine cellars underneath the town.
11) The POZALAGUA caves, in the Karrantza valley, a spectacular combination of rare stalactites and stalagmites, now specially prepared for visits with kids, http://www.karrantza.com/?seccion=cuevas3&idioma=es.
You will not find massive tourism or tourist traps at any of these places, and if you plan your visit in advance you will get a very good idea of this beautiful country.