Explore Montrose and the Museum District--they are vibrant areas of Houston filled with all sorts of gems.
Go to the Menil Collection, spend time in the nearby park, and spend some time reflecting in the Rothko Chapel. All are free and within walking distance of each other. The Contemporary Arts Museum is nearby and is also free.
Make a stop at the Chocolate Bar to indulge yourself with decadent chocolate ice cream. La Dolce Vita is the place for artisanal pizzas and has a great wine list. Westheimer is filled with boutiques and resale shops and trendy cafes.