Visit the Bangkok National Museum located just next to Sanam Luang near the Grand Palace.
You can get there easily by taxi, but the far more fun way is to travel along the "River of Kings" by blue-flag tourist boat (50 baht) or orange-flag ordinary boat (14 baht).
The National Museum is located just behind the Grand Palace on the grounds of what was once the "Front Palace". Long ago, in the early decades of the Bangkok Kingdom, the King of Siam would appoint an "Uparaja", or "Second King" to serve as his viceroy and heir.
The seat of the Uparaja was the Front Palace. This changed in the late 19th century with reform to the rules of succession, and the Front Palace was eventually converted into the Bangkok National Museum. Now, the National Museum houses an impressive collection of Southeast Asian art.
While the exhibits are all indoors in air-conditioned rooms, you will have to walk outside to get between the different buildings of the old palace complex which now makes up the museum. It's a nice place to visit on a rainy day, but you might still want to have an umbrella with you!
Admission to the museum is 200 baht. A visit to the National Museum is an excellent way to spend an afternoon after a morning sweltering in the heat at the Grand Palace.