Drive in the daytime only - there are unmarked culverts that can be a disaster to encounter in the dark. I would not worry about personal safety other than driving safety - our roads are very slick in the rain, and you need to watch carefully for kids, animals and other drivers. Don't miss the Belize Zoo, take a quick turn around Belmpoan, maybe drop in at the Inland Blue Hole. San Ignacio, River trip, Cahal Pech ruin and museum.
A road trip south is also fun - rising rivers near Kendall could provide problems though - check for conditions before you set out.
Generally you would be leaving from Belize City and returning there - making the car thing easy but you also have the option of renting at Budget in Belize City and returning the car to Placencia.