Climbers flower better on horizontal stems. If yours are too stiff now to train horizontally try cutting lower down above a bud if you can see one and as the new growth appears try to train it either across a wire if on a fence or round a support if its on an arch for example. If this makes you nervous try cutting just some of the stems and do the rest next year when you are more confident! In any case cut off the smaller sprays that flowered last year if you can reach them somehow, and trim back the ends of the old stems if you are going to leave any. Flowers will appear on the current year's growth. There is no need to remove any remaining old leaves but if there is any sign of black spot on them do remove those and burn them, also picking up any on the ground so the new growth isn't infected. If you put up a photo of the flowers in the summer someone on here might be able to identify them for you.

If it has several stems take out the oldest one or two to the base, this will encourage new shoots that you can train horizontally. Tie in new shoots so that sideshoots come all along them, with flowers later. Keep taking out big old stems each year and you will encourage flowers lower down where you can see, smell and pick them!
Roses are tough - don't be afraid to prune!