You can get Information on Italy and several specific towns here on this website. Also click on the tour information to see what is highlighted in the tours.
I'm guessing that you may not have been to Europe in a long time. Because of that, I also recommend the following RS links.
Money. Remember that if you make reservations for anything ahead of time from the US, inform your CC company that you are going to do that and where so that your card will not be rejected. Also, whatever resources you use, set them to € (EUR, euros) so you are getting a true picture of costs.
Packing Light. Especially this Travel Talk video by Sarah Murdoch. And the Packing forum. Even though you plan to stay in apartments, you don't want to be lugging heavy luggage on trains or buses or boats to get to them. I'm 72 and I travel with very little compared to my husband.
I agree that staying outside the cities you want to go is not the best idea. For me it's much better to be within reasonable walking, bus, metro or taxi distance of the sites I want to see and the food I want to eat. I like to stay in apartments or B&Bs, not big hotels.
My favorite resource to find those is Booking.com. I pay attention only to the review score, no matter what kind of lodging I'm seeking. Star studded amenities aren't high on my wish list. Apartments and B&Bs usually do not have stars anyway. Use the opening screen's search box to do the basics. Then filter them according to your needs. There is a learning curve. Most apartments will have a washing machine, though a dryer will likely be lines out the windows or a rack inside. Pick a washing machine as a filter for your apartment searches.
There are many other resources for finding apartments. Owners sometimes list their rentals on several listings services. The prices may vary from one to another. Often you can also find the same place listed independently or contact the owner and rent directly through them.
With 30 days, you could do a round trip to one airport and make a big circle. Or you could fly into Milan or Rome or Venice or even Sicily and fly home from a different city. Here's where the part about deciding where you want to go based on what you want to do or see there is important. Nonstop flights are preferable. If there are none from NOLA, Houston and Atlanta would both possible short hops to catch one.
My favorite resource for finding flights is Google Flights. You can set up price tracking options, see pricing history, etc. When you decide on something, you go to the airline to make the reservations.
And if you plan ahead well enough, you can get really cheap train tickets. I like to use Trenitalia. I love taking the fast Frecce (arrow) trains. They're great for covering long distances quickly, although I probably wouldn't take a train to Sicily unless I was already in southern Italy. I'd fly. Google Flights covers intra-European flights, too.
Have fun with the planning and make sure everyone is involved.