The third alternative to get into town from the station is the bus. The cross-town trolley bus runs every 15 minutes, and passes all along the seafront the other (west) side of the centre, where most (but not all) of the hotels are. You want one going from the train station side of the main road outside, with the destination board reading Villa Helios.
There are tunnels all along the line from Nice, mostly reasonably short, but there is a longish one between Nice and Villefranche, a long one under Monaco with the station in the middle, and the San Remo tunnel, which is nine miles long with station half-way. Between the tunnels (the vast majority of the way) the views are excellent. Train travel in these tunnels is a great deal less claustrophobic than the London Underground.
If you wished to avoid the San Remo tunnel, you could take a bus from Ventimiglia. These run every 15 minutes during the day, take 40 minutes and end up in the centre of San Remo. They only cost €1.50 (tickets bought in advance at Ventimiglia station bar) or €2.50 on the bus, but are not very luggage friendly. Same price for the San Remo cross-town bus.