Artfahie,
Cycling shoes have a much stiffer sole than regular shoes. This helps apply more power to the peddles more effectively and efficiently. Are they required? No. Are they recommended? Yes, I believe they are. Do you need to spend a lot of money? No. Cycling shoes will help you enjoy cycling more because you will get better at it the more you cycle and your equipment will support that...and that feeds the desire to spend more and more time riding.
I have a dedicated pair of ~25 year old Diadora road cycling shoes than I wear with my Look pedals but I don't use those when commuting to work. I wear a pair of Shimano MTB shoes for my commute and any ride on my fixie or just going around town. The cool thing about them is I can wear them around off the bike. Would I walk a mile in them? No. But for beating around they work fine. I got them at REI. They are bright blue and were less than $100. High quality shoes.
I would get cycling shoes before a buying a jersey but after buying some padded cycling shorts. Cycling shorts, in my opinion makes a HUGE difference. Cycling shoes rank right up there though. The next progression would be toe-clips. Once you go there you'll be hooked for life. You have been warned.