If you're flying into IAH, you can take a Metro bus into Downtown or a cab (about $55). If you're flying into Hobby, then you'll need to take a cab (about $45 to downtown).
Head downtown and visit Discovery Green park and The Heritage Society. The Grove is a lovely restaurant inside the park that would be a good lunch spot. You could also explore downtown by downloading audio walking tours--it's a really cool way to learn about the city. From downtown, you can also take the light rail and visit the Museum District and Hermann Park. There are so many lovely places there, including the Museum of Fine Arts, Houston Museum of Natural Science, The Menil Collection and more.
Or if you like to shop, take a cab (about $35) to The Galleria. You'll find 375 shops, an ice rink and an entire wing of great restaurants. Less than a mile away are two more shopping centers, Uptown Park and Highland Village.
If you want to stay near IAH, there is the National Museum of Funeral History. It's a quirky place whose motto is "everyday above ground is a good one."
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