Time is Your Biggest Constraint
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Lizzie and Tessa: As with any major RoadTrip, the time you have available for making it is your major limitation - often even more so than cost! As fuzzypuppy pointed out, it's a solid 2½-3 days from New York to New Orleans even if you don't stop at interesting attractions along the way or take any 'roads less traveled'. So, let's first concentrate on what you can do if you only have three days for the one-way drive.
With three days you can either take the direct route and make 2 or 3 short stops to see something. Or you can take slightly slower/roads to enjoy something than just Interstates. The former would entail using I-78/I-81/I-75/I-59 through Harrisburg (PA), Knoxville (TN), and Birmingham (AL) with stops such as Shenandoah National Park, Chickamauga and Chattanooga National Military Park, and maybe Huntsville (AL). The later option might include using US-11 as an alternative to I-81 or even using the Natchez Trace Parkway between Nashville and I-55.
If you can devote another day (or more) to the drive, then the possibilities really start to open up and you can pretty much pick any sites/roads that are even 'more or less' between New York and New Orleans and get to at least some of them on a very relaxed journey. So, the first things you have to decide about any RoadTrip are: How much time do I have? and What (sort of things) do I want to see? Once you've made those decisions, we can be a lot more help with specific recommendations
AZBuck