The Louisiana purchase doubled the size of the USA and was, undoubtedly, a pivotal moment in the history of the country. The agreement was signed in a room on the second floor of the New Orleans Cabildo, which you can visit for a nominal fee. The Nola Cabildo is also the oldest government building in the country and is right next door to the Nola cathedral in the heart of the French quarter so you will almost certainly pass by it anyway. Definitely head inside and explore what the building (now a museum) has to offer. In the summer it provides a nice relief from the hot sun outside whilst still being able to hear the great jazz coming in from the square outside.