What is the best way to get rid of a mold/mildew problem? The garage apartment that we have has a mold problem that just seems to ruin the walls. There is no exhaust fan anywhere and the whole apartment only has 4 windows. I'm sure there is no water trapped inside the walls but I do have my doubts. Should the sheetrock be replaced or can it be saved?

There needs to be a ridge vent on the roof, soffit vents so it has air to draw from. No insulation over the soffits. If the sheet rock is now all soft then it needs to go.
Any sheet rock that gets replaced needs to be paperless sheetrock.

To understand why you have mold you must first understand why and how it grows.
Mold needs three things to grow. A food source, oxygen and moisture. We cannot control the food source or the oxygen, but we can control the moisture.
With that said. removal of the mold is the easy part. Controling the moisture that caused the mold can be somewhat difficult.
To start with. If you have mold growth on anything other then cement, glass or metal. The material must be removed and disposed of. Any wood framing that cannot be removed must be sanded clean, all visible sides, then painted with a anti-microbial encapsulant paint. The cause of the moisture must be determined and corrected. Once that is done, rebuilding can take place.
Think of mold like a weed in the lawn. It has a root system that cannot be seen and the flora which grows on the surface like a blade of grass.
Simply cutting the grass off at the soil level leaves the roots for the grass to grow back. The same holds true for mold. Cleaning the surface of the sheetrock with a cleaner and repainting it leaves the roots intact. You would have to soak the walls until saturated with cleaner to kill the entire mold growth. By then you have distroyed the wall. You must remove the rock completely and follow my suggestions to get rid of it for good..
But remember if you do not control the moisture within the walls the mold will come back even with new sheetrock on the walls.

Also because the walls in this apartement have mold on them. It is most likely that the mold growth which appears black? or very dark Green is stuff you do not want to play around with. Be sure to wear protective gloves, eye protection and a breathing mask. You do not want to breath this stuff in. Also the room should be sealed off from the rest of the home with plastic. A vent fan system should also be installed to prevent the mold spores from spreading to other areas.
I know this sounds like overkill, but being involved in this industry and seeing what I have over the past 12 years you would be amazed how easy you can get sick while removing this stuff.
My suggestion is to have a professional mold remediation contractor remove and clean up the area. Be sure to have pre and post air tests done to be assured that they did the job correctly. Only then can you be assured that its safe to re-occupy the room(s).