Honeysuckle climbers are relatively easy to prune - the most important factor being correct timing. If you have the space for them to grow into large climbers, then pruning should only be required by way of trimming back to fit the space that you have to fill.
If your space is restricted - or you want to keep it under some sort of orderly control, then most climbing honeysuckle vines can be cut back quite hard in the late winter or very early spring each year. They will then flower on the new growths sent up in that season.
These new growths - vines - will often reach 2metres in length, so it will re-cover a good size trellis during the year.
One or two varieties of Climbing Lonicera are best cut back right after flowering in the mid to late summer. These are members of the Lonicera periclymenum group, and are some of the most popular varieties in the UK. They include (pictured) Lonicera periclymenum Serotina - the late Dutch Honeysuckle; Lonicera p. Belgica - the early Dutch Honeysuckle; Lonicera p. Graham Thomas.
These varieties tend to flower on growths - wood - made the previous year, so should be cut back after flowering - to size determined by the available space. In any event they should be cut to just above a strong new growth that will be sprouting from below the flowered area.
The evergreen Lonicera japonica types - including Lonicera japonica Halliana - do not actually require pruning - other than trimming to size. Again this should be done in early spring. Summer is too late - you will lose the flowers for a year.