Sanremo has a good café culture — this is northern Italy, and the espresso is taken seriously. The main café strip is along Corso Matteotti, the pedestrian shopping street that cuts through the centre of town. There are also several outdoor bars on the seafront promenade.
The casino has its own bar and is worth visiting for a drink even if you are not gambling — the Belle Époque interior makes any drink feel more significant than it has any right to be.