Well, there are lots of nice parks in all our cities. Guilford Courthourse National Military Park in Greensboro is a nice park with lots of history. Part of the Revolutionary War took place there. It is also connected to a couple other parks.
Winston-Salem has Old Salem, which was founded and settled by German Moravians. This is close to downtown. Lots of colonial history there with cobblestone streets and late 1700's/early 1800's buildings. Winkler Bakery is a great place to get some goodies like Moravian cookies and sugar cake.
Hope this helps.