Day trips around Sanremo.

Heading west you have Ventigmiglia (large market and a very good botanical garden), then in France you encounter Menton (multiple gardens and medieval center), Monte Carlo, Beaulieu-sur-Mer (for the Villa Ephrussi de Rothschild/gardens and Villa Kerylos), Villefranche-sur-Mer and Nice. Travel time to Nice is 84 minutes; the others would be faster. You could also hop off the train at Eze-sur-Mer and take the bus up to Eze Village. There are also lots of good spots beyond Nice, but you'd be getting into a lot of travel time then.

Heading east there are many coastal destinations that sound lovely, but I haven't been to that part of Liguria (or to Ventimiglia). I'd research Dolceacqua, Apricale, Bordighera and Savona among the places closest to Sanremo. Savona's said to have a lot of Art Nouveau architecture. There are some interesting-sounding places up in the hills, too, some with bus service from Ventimiglia.

You'll easily be able to keep yourself occupied.

Well, a car gives you more options up in the hills, but it wouldn't necessarily be a benefit for coastal locations. Parking can be difficult (and likely not free), and traffic along the coastal road is subject to (unpredictable, I was told) significant delays.

Edited to add: I meant the coastal road in France. I haven't been across the border.