I'm glad you asked. Let's go through some common do's and don'ts of swing dance footwear for beginners.
Shoes for Everyday
What you're looking for is something comfortable that doesn't have too much tread on the bottom. These are usually casual sneakers, not super athletic shoes. Benny from The Sandlot would have made a great swing dancer; P.F. Flyers are a perfect swing shoe. Listed are a few of the most common brands I see out on the dance floor.
I dance in Toms, day-to-day. I find them to be the right amount of slickness for most floors and they are by far the most supportive tiny shoe I've used for dancing. Their canvas upper will stretch a little as you wear them, but not too much. I have wide feet and dance many, many hours a week, so I buy them snug. Toms run about $60 a pair, which was a hurdle for me for a long time. If that's more than you're willing to put down right now, I suggest a trip to Payless. I danced in shoes from Payless for many years. Find yourself some knock-off Keds for $15 and wear them until you burn a hole through the sole. Literally. They're not great on the feet, but they are easy on the wallet.
The kind of shoes you get are not important. What's important is that they're comfortable, that you like them, and that they serve you well while dancing. Use this a a guide, not a rule. By no means do you need to wear black shoes. That was a coincidence.
Some people like to purchase dance shoes with sueded bottoms for extra slide on the dance floor. I think that's a fine choice for those with a little experience. A place like Aris Allen's makes them for swing dancers. Some of their shoes are available in New Orleans at Trashy Diva, but not all of them. You can find them online at dancestore.com. If you're just starting out, you don't need to take this step yet. You know you're really committed to a hobby when you buy the shoes. Give it some time.
I'd just like to point out that you can also take your regular shoes to the cobbler to have them sueded. A good cobbler is your best friend. I go to Wilson's Shoe Repair at 3837 Frenchmen St.
Things to Avoid
Try to avoid heavy shoes like winter boots, shoes that come off easily like ballet flats, or anything open-toed like flip flops. Heavy shoes are tough to move around in and hurt more when used to step on others' toes. Flimsy shoes hurt more if you're the one getting stepped on. Toe steppage is inevitable.
Also, keep your sueded shoes for the dance floor only. Wear something else to the venue and bring a shoe bag.
The world of fancy dance shoes is robust and beautiful. I could (and will) spend a lot of time cataloging dress shoe makers for swing dancers, but for right now let's stick with the basic concepts.
Oxfords and Loafers
Leather-bottomed dress shoes are the way to go! They look great, they're durable, and you can take them to your best friend the cobbler for repairs. This is a good investment for a serious swing dancer. Start with a dress shoe you find at a department store or a shoe store in the mall and then work your way up to an Allen Edmonds or Frye oxford.
Keep it short and keep it comfy. Start with an Aris Allen mesh Mary Jane. This is the starting swing dance heel if there ever was one. Work your way up to something from Remix. Go to Trashy Diva and start trying things on.
A note about heels: wear them or don't wear them, there's no rule about it. Some people really love them and exclusively dance in them. I wear them for balboa, for performance, and when I'm dressing up nice. They take some getting used to. I recommend breaking them in at home before going out dancing, especially if you're not used to wearing heels.
things to avoid
Watch out for shoes that have a leather (or often fake leather) upper and a rubber sole. They can be deceptive. Make sure to check! That said, I don't mind if the heel, or portion of the heel, is rubber. It can act as a break and I think works just fine.
Expect more on dress shoes and heels in the future; that's a whole other world that I can't wait to tuck into. In the meantime, I hope this gave you a good start.
Comment with links to your favorite dance shoes for everyday, your favorite New Orleans cobbler, and questions. I want to know!
Love and Cobblers,