You can use any type of wood for carving if you have a dremel. If you are making patterns in wood, you might want to use woods that don't have a lot of grain, like basswood, maple or walnut. Other good woods for carving are cherry, myrtlewood, poplar, birch, or apple. I would stay away from porous woods like oak and hickory.
As for dyes, anything will work. They have dyes on the market that are good for just wood, but I find that boiled onion skins, beets, blueberries and most other colored fruits and vegetables will get you there, too. It all depends on what you want to do.