Mostly the same do's & don'ts as at home.
There have been some incidents in the Nassau area involving tourists on particular excursions. Last I heard the cruise lines had canceled those excursions. The disturbing part of these incidents is that they were armed robberies.
I'm not aware of any problems in Freeport, but that doesn't mean there aren't any. Just that I don't tend to follow the happenings in that port.
My general advice is stick to main thoroughfares in tourist areas. Stay aware of what is around you. If you see police it's a good thing. They are most likely there to protect tourists - not because it's a high crime area. Safety in numbers - lots of people around is generally good. Don't flash large amounts of cash. Ideally don't carry much cash. Don't wear expensive jewelry.
Downtown Nassau is a relatively small area with a good layout. It's pretty hard to get lost in. While it's not strictly a grid, the streets don't wind.
Have fun.