In terms of a climb like the first, the best training session I have found is hill repeats. Basically go out and ride up a hill that takes between 8-10 minutes at a 'sustainable' pace, then turn around at the top for a 4-5 min recovery and repeat the climb. Start off with 2 repeats and build it up from there, up to 6 if you really want to build the climbing. If not, do 2 or 3 intervals at a faster pace!
For the second, shorter/more explosive climb I think something like Hill Sprints would be beneficial. Say, 30 seconds flat out followed by 30 seconds recovery. And 10 repeats.
Other than that, I have become a fan of 'big gear climbing'. Pretty simple session to build strength. Find a hill around 10% and 400m long then ride it seated in the biggest gear possible then descend and repeat 5 times. Need a good warm up for this one and don't bother if your knees are suspect, it puts a lot of strain on them!
I guess it's fairly simple, comes down to power to weight ratio at the end of the day. A great excuse to spend more money on the bike to lighten it up
And I know how you feel with strava, good bye recovery and gentle rides