In reply to Kassius:
Getting out in what remains of the winter would be a good start. Given your fitness level and climbing ability there's no reason why you shouldn't get up some harder stuff than Striding Edge etc.
Regarding the Alps, you can go on a course if you like or hire a guide but there's no reason why you need to if you do your research and start on easy stuff or team up with someone more experienced. The technical difficulty of the peaks you mention won't be an issue, just stuff like crevasse avoidance and rescue, but you can read up on that, even if you do go on a course.
If you do hire a guide I'd recommend hiring a British one (arranged in advance) rather than a local one in Chamonix or Zermatt. They only seem interested in taking some money off you to drag you up the peak and back down as fast as possible rather than actually teaching you how to climb independantly.