How can I protect my feet during Zumba? Shoes good for Zumba?

To protect your feet during Zumba, first you need good shoes that support your foot, provide cushion, allow sliding and allow you to bend the sole of your foot.  Avoid heavy shoes, like those used for basketball, for instance.  Non-marking court safe shoes are important for many dance studio floors.
1) I do not recommend split sole jazz sneakers. Bloch Women's Boost Mesh Sneaker Dance Sneaker,Black,10.5 X(Medium) US   I started out with them but quickly realized they had no side support.  So when you trot sideways across the floor, you risk twisting your ankle.
2) I used to use Nike court shoes that are commonly used for hip hop.  They have a wide flat sole and a pivot point on the bottom.  I liked them for a long time, but they are a little bit stiff and don't allow maximum bending of the sole of your foot.  They also caused a lot of friction and callouses on my toes.
3) I do recommend Nike Musique. Nike Women's Musique IV Dance Shoes (7.5) But they are for narrower feet and will fit narrow feet snugly.  If you have wide feet, don't even bother.  They have arch support, a smooth sole for less friction on twisting and a pivot point.  They are quite flexible. They are marketed as a dance fitness shoe.  If you are thinking of ordering online, I take the same size in Nike Musique that I take in any other Nike shoe.  So you can try any Nike athletic shoe on in the store and order that size.  Beware- there are different styles of Nike Musiques.  Some have a little bit of mesh on the side.  Others seem to be all leather.  The all leather ones are smelly sweat chambers.  Go for mesh ventilation if possible.  I find these at DSW, Nordstram Rack clearance or on Ebay.
4) I also highly recommend Ryka Transitions. Ryka Women's Transition,Tyre Grey/Metallic Platinum/Metallic Baja Purple/White,10 M US
  They are for average or wider feet.  I have narrower feet and still wear them; they give me lots of toe room and keep the callouses away.  It's good to be able to lift your toes up and wiggle them while you dance.  You don't want scrunched up toes.  The Ryka Transitions have the most flexible soles, which is great for stretch at the end of class.  They also have fairly smooth bottoms and a pivot point.  Again, some have more mesh ventilation than other styles.  Look for the mesh.  I found mine at DSW.  Sizes do not run the same as Nikes.  I wear a half size bigger in Ryka.
5) I just tried on Ryka Influence and liked them a lot.  I bought a pair!  They have good support, cool colors, a smooth, flexible sole, a pivot point and good ventilation.  They have more cushioning than Ryka Transitions.  However, after my studio refinished its floors the surface of the Ryka sticks to the wood.  So I either need to use them with DanceSocks or save them for the tennis court now!
6) I also have two pairs of Zumba sneakers. Zumba Women's Z1 Dance Shoe,White,9 W US  They do the job but really don't have as much support as Nike or Ryka Influence.  They do have a lot of toe room and feel light.  However, they do not protect your feet well from heavy impact such as when you jump a lot.  If your zumba class is not real jumpy, these should be fine.  The surface of the Zumba brand shoes work great with the refinished wood floors.
7)  Due to the refinishing of our floors, I have been trying different shoes.  And I found Asics Noosafast that I think look great, are ultra-light and flexible and react well with my floors.
8) Should you run into sticky floors or carpeted surfaces, keep some shoe gliders such as DanceSocks in your bag.  They're cheap and will save you a lot of knee pain!

Here are some tips for your feet while you are dancing or even just walking:

1) Roll thru whole foot from heel to toe when you walk forward.  Even when you do a merengue march, roll through the whole foot.  When you take the backward step in the salsa basic, land on the ball of your foot and then settle into the heel.  This is why flip flops are bad for your feet.  People tend to walk flat in them instead of rolling through the foot.

2) Don't crash your foot into the floor.  stop your foot before it reaches the floor or as it reaches the floor so the impact  won't be hard and straining to your joints.  The stop engages your muscles also.

3) Dance Latin style with high arches (heels up)- this not only looks good and mimics the high heel look of Latin dancing but stretches your foot at the same time.

4) When you push into the floor with the ball of your foot to get grounded and more hip movement, relax and toes and raise them up.  Don't curl your toes inside your shoe.  That would stress a lot of muscles in your foot.  This is why it's good to have shoes with some toe room.

5) Incorporate foot stretching at the end of class.  For example, during a hip flexor stretch,  your back foot can have a very high arch, where you heel is all the way up and your toes can bend.

6)Get regular foot massages.  Or stand and roll your foot over a tennis ball to give yourself a massage.

7) When you're tired in class and you stop paying attention, that's when you can get hurt.  Always pay attention to what your feet are doing and know where you are putting them.

8) Don't try Zumba moves in high heels at a party.  Dancer use special dance high heels, not just street shoe high heels.