From Acqui Terme it is only 1 hour by car down to the Ligurian coast and a dip in the sea.
One of our favorite places is Finale Ligure, which is a little further west on the Italian Riviera towards France, by car it takes 1.5 hours to drive there.
Genoa is the closest big city to Acqui Terme, it is located in Liguria right where the "boot shaft" of the Italian turns west. Genoa has 550,000 inhabitants.
A really beautiful city where you can walk from piazza to piazza. Inside the city itself, there is a bit of a small town feel where you will also find smaller alleys and maybe you will find the street with all the palaces. Genoa also has a large port that is well worth a visit.
By car, it takes 1h 10min to Genoa from Acqui Terme. But there is also a direct train from Acqui Terme to Genoa which can be a very good alternative.