Tabarca Island was my favorite part of my trip to the southern part of the Alicante region of the Costa Blanca. The beaches in Alicante follow a general rule (in my opinion, at least): if you can reach them easily, they're not very good. In other words, the easy beaches are jam packed and overdeveloped, and the pristine hidden coves are on dirt roads outside the city. If you want that same virgen-beach feel but are traveling (like me) without a car, I'd recommend hopping one of the "Tabarquera" boats that leave from Santa Pola marina (15euros round trip) and heading to Isla de Tabarca.
Tabarca is a small island divided into two sections, a maritime village and a nature reserve, with a pleasant sandy beach and some restaurants occupying the middle part. This beach, while probably the most beautiful on the island, was packed when we went (as occurs in July and August). But, you can head to the natural side of the island and find some peaceful, uncrowded coves with sparkling emerald water, plenty of fish, and some neat little islands you can swim out to and explore. If you head towards the natural side of the island on the right side (the side towards the ocean, not the mainland), you'll find the most accessible coves about 50-100meters down. On the other side (the side that faces the mainland), the coves are rockier and more shallow, but also less crowded. In fact, there was only one other couple there with us, which says a lot considering that the main beach was crowded as hell.