Proper footwear is a good idea, but not necessary.

A good pair of flat shoes can give you a more stable platform. Light shoes can help you with things like running.

In terms of squatting, some people like to get a shoe with a slightly elevated heel to help going Ass To Grass for squats. Others prefer just socks or even barefeet.

Your personal preference will determine what footwear you use. Be aware shoes may save your toes every now and then.