You're not going to see a lot in a day trip. I think Ventimiglia is ~30 mins driving from Monaco, but it's going to be really crowded in August, and it's a pretty touristy place to begin with, in the summer. Avoid the pizza, Liguria isn't known for that - however, the focaccie from the region are pretty good. The pasta won't be what you're generally used to - heavier emphasis on seafood and filled pasta, some of which are served cold. These are features, not bugs, though, as the Italian food we commonly see in the US is generally an Italian-American representation of foods from the south of Italy. I still wouldn't go in August, though.
If you're going to leave Monaco for a day, I'd suggest you're better off heading to Nice, where at least the city is large enough to deal with the crowds.