You already have a great attitude towards it, that is the first hurdle.
Go with plastic pipe, ABS which is black, or PVC which is white. Then ask the guys at the plumbing store which glues or primers you may need. The plastic can be attached to the old plumbing with what is called a hub-less connector, usually goes by the brand name furn-co. Basically a piece of rubber like hose with some clamps to connect both different types of material.
As long as you draw a sketch of what you have there already to help along the way. Some changes you may need are called a single trap and some y and t fittings as you go, or post some pictures so we can tell you what to update.
Also check out your local library for some more plumbing books, they are free to borrow and there is usually more than one.
Someone will be along shortly to express more opinions.