I currently have a cheap set of regular pedals that can accept toe clips, but I don't use them. I looked at clipless pedals but can't figure out how they work. What are some brands of clipless pedals that I should look at? I do all road cycling, with some gravel/dirt riding but nothing that I'd consider mountain biking.

What kind of shoes do I need, and how do the cleats work?
Are toe clips better/easier to use?
Will I need to buy another pair of shoes just for cycling?

What happens when I am trying to find the side of the clipless pedal that accepts the cleats? Do I just spin the pedal? Wouldn't that slow me down each time I had to start up from a stop?