Golf shoes are a good idea, however, because they are engineered to keep your feet from sliding around during the golf swing.
Golf shoes have come a long way, stylistically, over the past 15 years or so, too. You can still find "old school" golf shoes - black and white wingtips, leather and tassels. But golf shoes now come in many styles and colors, some indistinguishable from other athletic shoes, some in moccasin and even sandal styles.
The comfort level of golf shoes is also much higher than it used to be. There are many different price points for golf shoes, including relatively cheap models. So if you are new to the game of golf, investing in golf shoes is a good idea.
Again, though, golf shoes are not a requirement to play golf.
So what if you don't own a pair of golf shoes, and don't want to spend the money for them? Then wear a pair of shoes with the most "gripping" soles you can find - rubber soles with lots of ridges, bumps, etc.