The red spot could be the result of dirty water , bacterial or parasitic infection.

How big is your tank? How many fish and how big are they? Have you done any partial water changes recently?  Do you have a water filter?

If you haven't done a partial water change in several weeks, i would do that first. This will help eliminate any type of water condition (such as ammonia build-up) that could be causing the red spot.

Have you noticed any unusual behavior from the fish recently, such as swimming erratically or scratching itself against some object in the tank such as a plant or rock? Is the fish eating normally ?

If you think the water quality in your fish tank is in excellent condition and you are not overfeeding your fish, then you can try a wide spectrum antibiotic treatment which you can buy from your local pet store.

black spot on the gill could be anything. use an antibacterial like Melafix and see if that clears it up. if it doesn't, post again with a picture.
changing the water is something that should happen normally, not a mark of a dying fish. yes, it should help clear up the spot if the spot is bad. you should consider your fish uncertainly sick but sick nonetheless and treat it as though it were mortally ill and about to bottom up. being paranoid about your fish isn't always such a bad thing; if you watch your fish avidly for potential problems and setbacks and dangerous things but know in your mind that they are most likely ok and happy, then you will have a very happy existence with your fish. in essence, scan carefully, like you were looking for a problem (but don't look and force yourself to see a problem that doesn't exist) and understand that when problems happen, there are a bunch of people who have had the same thing, and there are also other people who can tell you how to fix it.