In reply to Taurig: I really like your outlook
There is nothing better than being able to climb classic routes like Tower Ridge efficiently and quickly using the full array of climbing and mountaineering skills. A lot of climbers I know, really struggle doing anything other than basic pitched climbing with standard rock climbing belays.
Lots of good advice, but I would make a couple of comments/suggestions.
First, a quick bit of sage advice: If the weather is at all dodgy, do not go scrambling, go walking or proper climbing. You have infinitely more chance of coming away unscathed if you fall-off a wet VDiff than if you have a slip on a wet Grade 2 scramble. Scrambling should be viewed as very much a dry weather activity.
Second, don't overly worry about slings for belays - you've got the rope for that.
Third, 95% of UK climbers and mountaineers carry far too much kit.
- Partly this is due to the marketing efforts of the gear manufacturers who persuade you to buy 'full sets' of wires or hexes that you don't really need. You can easily get away with wires in sizes 3-9 and 2 medium hexes (plus possibly a large hex or a size 3 cam) for the vast majority of both scrambling and easy climbing.
- The same applies with quickdraws. The relative demise of 'nuts on cord' and small hexes combined with the increase in sport climbing (and lighter gear) which has led to an explosion in the number of quickdraws climbers carry. A really basic rack in the 1980's only had 2 'extensions' where as many climbers now think 6 is on the low side and often recommend 8.
- Finally, people carry far too many screwgates. This a legacy of the climbing community failing to differentiate between how climbing instructors are taught to build belays when taking school kids climbing where the belay can be in use and out of sight of the instructor for a whole day and what is an appropriate belay when mountaineering.
In all these areas, people generally buy a fairly large amount of gear so get into the mindset that they actually need it all, which is rarely the case.
By all means go out and buy a full set of nuts on wire, a set of hexes and a set of quickdraws (it is eminently sensible and far cheaper to do it that way) but avoid falling into the trap of thinking you need to always carry everything.
Fourth. Seriously consider hiring a Mountaineering Instructor or better yet, a Mountain Guide for a day or two of scrambling/mountaineering training. When I've done some training with Guides I have learnt a phenomenal amount. It is very worthwhile and you will pick up things far, far quicker than you will just by reading about it and trial and error.
Finally, whatever you do, have fun.