You can just take green wood cuttings and insert them in a pot of well drained compost. Cover with a plastic bag for several weeks, keep out of direct sunlight, and check that they aren't rotting. They should root in 3 weeks to a month, and when you notice them picking up, remove the plastic bag and keep watered.
A much easier and more foolproof way is to layer the honeysuckle. Take a leading shoot, without flowers, and bend it down in the middle into a pot of gritty compost, with the bend below soil level in the pot. It helps if you slightly damage the stem (nick it carefully with a sharp knife, not to sever it) as this encourages root formation. Put a stone on top of the pot or a large staple to keep the layer in place.
After a few weeks or months, you should see roots coming out of the pot, and at that point you can sever the layer from the parent plant.
Don't forget about it and keep the pot watered regularly to ensure rooting.