What is the best tool/method for vine removal?
In addition to the normal tools like pruners, hedge shears, etc., my tools for controlling unruly vegetation include:


pruning saw (look for one that cuts on both push and pull strokes):





bypass lopper (much more leverage than a regular pruner):





tree pruner (no more climbing on a ladder to get at high branches!):




In your situation, I would start by using these tools to cut back all the vines close to the ground.  Dig as much of the roots out as possible, then keep an eye on the area for new shoots and dig them out (or spray them with weedkiller) as they pop up.  Eventually you'll get rid of them, though it may take persistence on your part.