Most of my cycling is in traffic unfortunately I don't get any foot ache. Part of me thinks that I am perhaps not peddling correctly as I think I am not putting the correct part of my foot onto the pedal. The first 20km are always hard afterwards I am ok. I wear running trainers for cycling
If you're peddling, you're definitely doing it wrong, you should be pedalling
Seriously, my commute for many years had 32 sets of traffic lights each way, but I wouldn't have considered not using clipless pedals and shoes. My favourites were speedplay frogs, but they don't sell them any more. Any decent mtb pedal and mtb shoes will do and will work well for the longer runs too, and you won't walk like a duck at the cafe stop. Road pedals and shoes are only really worthwhile if you intend racing.
Running trainers are not ideal as the soles are too soft to support the feet on the pedals properly, unless the pedals have quite a large platform. Cleats make a big difference, when climbing particularly.