What you replace the carbon with depends on what type of filter you currently have. It's easier to fit certain types of media into different filters.
If you are cycling without a fish, don't do any water changes until all of your ammonia and nitrites are at 0 and your nitrates are high. That will mean that your cycle is complete, and the water change will bring the nitrates back down. If you do a water change before you have nitrates, or while you have some nitrates but the ammonia and/or nitrites are still high, you will remove them from the water and the cycle won't complete as fast.
If you are cycling with a fish, do any water changes necessary to keep the ammonia below .25ppm and the nitrites as close to 0 as possible. This will take longer, because you're removing the bacteria's food! If you don't do these water changes with a fish, though, you risk hurting or killing it.
I cycled my last 5 gallon in about 3 weeks. I was using Nutrafin Cycle, which may or may not have helped the cycle, depending on whether you believe in those bacteria additives. Generally, it will take anywhere from 6 - 12 weeks without a fish, and longer with one because of the necessary water changes.