The suggested routes below are, in my opinion, those places that you should visit when coming to this so much 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.
3) ARANTZAZU Sanctuary (www.arantzazu.org) and OÑATI and its University(www.oinati.org).
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 and perfect beaches for surfing.
5) BILBAO, Guggenheim Museum (this was obvious!), Old Town (Casco Viejo), Funicular train to Artxanda for spectacular views of the city.
6) The 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 guided visit to the spectacular restoration of its Old Cathedral (Ken Follett based its novel World Without End 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,, https://www.karrantza.com/?seccion=cuevas3&idioma=es.
You will not find massive tourism or tourist traps at any of these places, and distances from one to the other are not that big, so plan your visit in advance and you will make the most of this absolutely beautiful corner of the world!!