Carbon is a major resource needed in a lot of Slimefun recipes. I found that the easiest way to get carbon, starting out, is by using Industrial miners. Using the upgraded version, which is cheap and consists of 2 diamond blocks, 2 pistons, a blast furnace, and a chest, (these are all vanilla items). This will give you a bunch of ores (between100-300) usually, The higher the Y level, the more ores you will receive due to the fact the there is more ground underneath you. Make sure you have some sort of fortune on your picaxe and mine all the diamond ore. You can then place all those diamonds into an Ore grinder and get carbon. 1 diamond will give you 4 carbon. The Industrial miners will also give you a lot of coal in addition, which can also be turned into carbon.
The fastest way to get carbon would be to make a wither skeleton farm. This produces coal, which can be turned into carbon.
I recommend saving a lot of what you collect, you really don't know what could end up being apart of a recipe. It's better to have it and not need it, than to need it and not have it.
If you have any questions about making other things easier. Don't hesitate to let us know, we are always here to help anyone and everyone.