Uprooting climbers

I was checking around my garden today and found a few ivy plants growing below the ivy bush that I didn't even know about. One of them had layered itself and grown some powerful roots and another has grown around the base of my tree which obviously needs remedying. Also there are a few babies just sprouting from the ground.
As I want to grow ivy or any climber up the fence I want to pull the ivy up, or split the roots in the case of the one that layered, and plant them where I want them. Are there any pitfalls I should avoid when doing it? Anything I should know first because I've never done this before?
I'll likely use the one strangling my tree as a practise.