Honeysuckle vines are easy to start by several methods. They can can be started by layering. (Takes the longest but gets good results.) This is done in the early spring when the shoots on the mother plant are flexible and dormant. Choose a young, flexible stem from the mother plant that is long enough to bend down to the ground a bit away from the mother plant. Pin down the shoot about six to nine inches from the the tip of the branch. Bend the branch into a "U" shape and dig a trench in which to put the branch so that you can cover it with soil between the pinned-down part to near the tip of the shoot. Leave the tip above ground. Once the shoot starts to grow, it is helpful to place a stake next to it and attach the new plant. If you start this procedure in the spring, your shoot should have formed a good root system by the end of the summer. At this time, you can cut it away from the mother plant and plant it. You can also leave it attached to the mother plant until early the following spring before it starts to grow. Then just clip it away from the mother plant and you have a new plant.
You can also start honeysuckle vines from cuttings. Cut a length of green, softwood growth from the end of one (or several) of the vines, making sure to get several sets of leaves. Strip the leaves from the end of the cutting nearest the cut end. You should have one or two leaf nodes bare and one or two sets of leaves left on the vine. At this point you have a couple of options. One method is to dip the cutting in rooting hormone and place it in damp potting soil or other rooting medium. The other method is to place the cutting in a vase of water and allow the roots to develop. In about one to two weeks roots will begin to grow. When you have several roots that are an inch or so long, you can plant the new plant in a pot or in the garden after danger of frost has passed.