Most folks know that when they sign up for a race - especially a race in which they know they will be pushing themselves and bodies to their limits in order to perform their very best - some soreness is going to be involved. But general post-race muscle soreness (and subsequently, how to prevent it) is a little different from the foot soreness you are asking about. The first thing you need to know and consider is really how often your feet are getting sore, and the level or extremity of that soreness. Grueling foot pain and soreness after a run is not normal, and might be an indicator of something wrong that needs medical attention.
The first thing to do is stretch. Yep, stretching your feet might not be your first thought but it is not actually as difficult to do as you might think. Get a roller or massage ball (a golf ball or tennis ball also works well and are about the right size) and roll the arch of your foot over the ball, allowing the ball to massage that pressure point just under the arch.
A hot foot bath with epsom salt will also help not just relax your feet, but the salt will help reduce inflammation and swelling. If they are really sore, an ice bath (and going back and forth between hot and cold) will also reduce inflammation and reduce pain. To help avoid inflammation before your run altogether, focus on your diet and make sure you are getting in some key anti-inflammatory foods. Finally, consider getting new shoes. Shoes that have been worn and run in for too long lose their cushioning and support, and can leave your feet extremely sore.
If the soreness is more on the outside of your foot around the skin, be sure you are wearing appropriate socks (thick ones are good) to reduce blisters. Incorporate a regular skin treatment routine into your day as well, that includes exfoliation of your feet (to remove dead skin and help reduce calluses) and moisturizer.