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Charlestown is a very attractive and historic Caribbean town, which can easily be explored on foot, because it is not large at all. True, there can be a fair bit of traffic on the mornings of market days (Tuesdays, Thursdays, and Saturdays), the roads are narrow, and there are sidewalks only in the middle of town, but cars and vans are generally not moving fast, and so if you keep well to the side of the road, and use reasonable caution crossing streets, you should be just fine walking all around town.
In the middle of the day the tropical sun can be quite intense, and it can feel quite hot in Charlestown then, so if you plan to explore a fair amount on foot, you may want to go early in the morning, or later on in the afternoon. The shops are open from 8 am to 4 pm weekdays, a half day on Saturday, and closed on Sundays.
If you want a very short but attractive walk, you could start at the Alexander Hamilton House/Museum of Nevis History, and simply walk south a few blocks, past the recently restored Nevis Tourism Authority building, through Walwyn Square, and over to the Courthouse and Library building on Memorial Square. Then you might like to stroll down Prince William Street to the beach at the north end of Gallows Bay, where the fishermen pull up their brightly-painted handmade boats in the morning and sell their colorful catch of tropical fish fresh from the sea.