You can play in cros-training shoes or service-worker athletic shoes with the nubby or shallow-tread soles. These work best on par 3 courses, or rather flat courses. If you play on hills, spiked golf shoes are a real help.
This is not just about missing shots - you can pull a thigh muscle or sprain a knee if you slip funny on a swing. A friend of mine took lessons and the first time we played, he showed up in street shoes (leather-soled oxfords). He slipped a couple of times on hillside shots.