Dancers have several options when it comes to footwear:
Regular socks: totally fine for class, but on pain of death, do not wear this to a milonga.
Dance socks: $5ish over-the-shoe dance socks (yep that's what they're called) that can make any shoe easily pivotable. The brand does not matter as long as the sock is shaped in a loop. Again, great for class but not as presentable for milongas.
Ballroom/latin shoes: as long as they are comfortable! If it is hard to pivot in them, just step into some baby powder or loop a dance sock over the front. If they are too slippery, just use a wire brush to rough up the soles.
Buying new shoes: We recommend ordering from sellers that offer free returns so you can try them out.
There are many beginner options under $40 for men and women, so don't break your bank for the first pair because you will learn what features you like/dislike over time. At this stage, shoes do not need to be specific for tango, and can also be found by searching for "ballroom shoes", "latin dance shoes" etc.
Leaders' dance shoes:
Any lace-up with a suede or leather sole.
Black colors and a glossier upper, as opposed to suede, make cleaning easier.
Some shoes are flat and some have up to a 1 inch block heel; both are fine.
Followers' dance heels:
A strap at the ankle is needed to stop you from stepping out of the shoe.
Styles that are open at the toes and closed at the heel are easier to maneuver.
Avoid block heels as they can be hard to pivot. Flared heels can be more stable than stilettos.
If this will be your first pair of dance heels, we do not recommend going higher than 3 inches.
Unisex practice shoes:
Worn by both men and women to classes and practicas, but not so much for milongas.
Can range from jazz oxfords (more affordable), or tango boots (more expensive).
For class, you are welcome to wear all comfortable clothing.
For milongas, common attires include:
Leaders:
Button-up shirts with a collar
Slacks
Some more advanced dancers add suit vests and blazers as well
Followers:
Sleeveless tops
Flared skirts or dresses
For both roles, it is recommended that you AVOID:
Tops that can fall off shoulders
Accessories or clasps that can get caught in closed embrace
Dresses/skirts that are too long
Pants that are too wide or have turned-up cuffs that can be caught by heels
Clothing that is not breathable