Running can definitely be hard on your feet. In many cases, you can keep your feet healthy by wearing shoes that fit, wearing good socks, and taking time off if you develop blisters or other sore spots. However, there are times when running gets the best of you and your feet could use some medical care. See a podiatrist if you're having any of these issues as a runner.
Lingering arch pain.
It's not unusual to experience some arch pain after a long run or a really hard workout. If this happens now and then, it's probably just muscle soreness. However, if the pain lingers for more than a couple of days and takes on more of a "pulling" quality, then it is more likely due to plantar fasciitis. This is an inflammation of the plantar fascia, a band of connective tissue that runs along the bottom of the arch of the foot. It can be a really nagging, lingering injury in runners, but if you address it early on, you can get ahead of it. See a foot doctor and they can recommend stretches and strengthening exercises that should help it heal faster.
Unless you're running ultramarathons, black toenails are not something you should be dealing with as a runner. They generally mean your shoes are pressing too hard on the tops of your toes. The black color is actually blood seeping out under the nail. It's a good idea to see a podiatrist about this problem for two reasons. First, they can poke a hole in the nail and safely drain the blood, which will greatly reduce the amount of pain you're feeling. Second, they can look at your shoes and recommend a different pair that won't put as much pressure on your toes, preventing this from happening again in the future.
A localized pain in the top of your foot.
If you ever develop pain on the top of your foot, and it feels very sharp when you press on a specific area, you may have a stress fracture. This is a crack in the bone that usually develops after a big increase in mileage or months of hard running. A podiatrist can take x-rays to diagnose the fracture, and if needed, they will give you a boot or soft cast to walk in.
If you're struggling with any of the problems above, don't hesitate to see a foot doctor. They can be really helpful professionals to have in your corner when you're a runner.