This area of Sardinia is pretty much apart from the other regions in the island. Some would say it is more touristic, hence less authentic. It is true that there is less of this "sardinian soul" that one could feel in more isolated corners. Historically indeed, because of its positioning and the relevance if its ports (Alghero, Porto Torres), the region has been more opened to the outside world, and was influenced by that. You will feel sometimes "just" in Italy, rather than in Sardinia. Yet this area should not be missed as it possesses a diversity of activities that would make any destination jealous. Looking at the sunset on Alghero from the Capo Caccia, enjoying the white sand of La Pelosa beach in Stintino, spending a day in a jeep to visit the Asinara island, driving on the coastal sinuous road between Alghero and Bosa, tasting a glass of wine in the tenuta Sella & Mosca ...all these will make your stay just unforgettable.
You will definitely want to spend a few days in the city. You will be charmed by the narrow streets in the old town, whose names are displayed in both official languages, italian and catalan. Alghero was in the past part of the Crown of Aragon, and you can feel this influence anywhere you go. Best way to quickly discover the old town is through the little touristic train ("Trenino") that you can request at the entrance of the port. There are many bars and restaurants; try to pick one that offers a view on the sunset. From Alghero, you can easily go and spend a day in the Capo Caccia (30 minutes away by car), with its spectacular cliffs, its Porto Conte Regional Park and the Grotta di Nettuno. The Grotta, one of the most famous marine caves in Italy, can be accessed either by boat (from Alghero) or through a stairway (Escala del Cabirol).
The road that links Alghero to Bosa (45 minutes drive) is one of the most beautiful on the island, as you will feel stuck between the mountains and the sea at all times. The best is to drive at sunset time, when colours are the most appealing. If you are hungry, you can stop by the panoramic restaurant that is 10 minutes away from Alghero. Also, there are along the road a couple of small white-sand creaks, that you can access if you park your car and walk for 10 to 15 minutes down the sea. There are very easy hikes, perfect for families with young kids as well. If you arrive in Bosa during the day, you will be impressed by the colours of the houses. Once you are in, you will feel like in a small province village.
From Alghero, you can take your car and decide to head up North, to Stintino, and its famous (yet a bit too touristic) beach. What I would recommend is to spend a day in the Asinara island. Asinara is mainly a National Park, housing some rare plant species and a fauna of wild horses, 200 donkeys (majority are albinos) and other mouflons. Besides, the island is also known as the "italian Alcatraz", as there was until the end of the 1990's two high-security prisons hosting famous detainees. The prisons were since closed, but you can visit them if you embark on a tour across the island. If you can afford, the best option is to book a personal tour in jeep, rather than join the touristic train. The quality of the explanations from the guide, and the freedom you have in choosing what and when to visit are worth it.
Because of its climate and the fertility of its soils, the region is known for the wineyards and the quality of the wine it produces (Cannonau, Vermentino). If you are an amateur, you will not want to miss a visit & tasting session in the biggest wine azienda of the region: Sella & Mosca.
For the sake of being a bit complete ... :-)