Think first about what you do when you get off the bike. If you do a lot of walking around, then consider MTB style pedals and shoes. Most MTB style shoes have inset cleats that make it eays to walk around, even on wood or ceramic tile floors, with slipping or damaging the floor. That's what I use. Youcan also find a lot of shoelace style shoes that lace up, which looks a little more vintage-y.

Road cleats work great if you don't walk around much, you can also buy covers for the cleats. The shoes tend to be more modern looking, and the pedals are larger, so there's that.

Most bikes use a standard 9/16" pedal shaft, so unless you're dealing with a french bike or a BMX, you should be OK.