Sanremo's best sights divide into two categories: the Belle Époque resort — casino, promenades, grand hotels — and the medieval old town of La Pigna, which is nearly the opposite in character. Spend a morning in the alleys of the old town and an evening on the waterfront and you will have seen the two faces of the place.
The town also rewards following specific interests: cycling (the coastal path and the Milan–Sanremo route), music history (the festival, the Ariston), and the curious footnote that one of the world's great scientific philanthropists chose to spend his last years and die here. Villa Nobel, Alfred Nobel's residence, is a genuinely worthwhile museum and quieter than the more obvious sights.