Use whatever shoe/cleat/pedal/whatever makes you happy. There's practically zero performance difference.
Typically people use road shoes with road cleats, or MTB shoes with MTB cleats. Road shoes with road cleats feels a bit odd to walk on, but it more or less works fine for short distances. MTB cleats sit recessed into MTB shoes, and walking like this feels pretty much like normal shoes. Road shoes with road cleats give you a wider, firmer/stiffer/whatever-feeling contact between foot and pedal that some people like, and they also tend to end up a bit lighter.
Cautionary note: putting 2-bolt MTB cleats on road shoes can work fine, but feels rather iffy to walk in, since the balls of your feet are contacting the ground via a narrow piece of metal.
...If you like platform pedals, they're also a perfectly legitimate option.