Definitely +1000 to what Tom said. His advice will definitely get you going, quite literally, in the right direction.
Really, mashing up hills is the way I built up, and continue to build strength. The hill right out of my driveway I generally mash up as hard as I can in the morning, but on off days I've been known to take it easy and spin my way up it
. As you get stronger, you will be able to judge this for yourself. But for now, embrace the burn. Love the burn. You'll have it around for a bit
With the shifting, again, what tom said. You have three rings, don't be afraid to use them! Figure out a good cadence for you, everyone is different in this respect!, and work on picking gearing to keep you in that range. If you want, use a bike computer with a cadence sensor, but I found that I personally learned much more about my body by just "feeling" what was right for me. By doing that for almost a year, I've gotten it down to a good rhythm, I can tell exactly how much effort and shifting needs to be done.
Also, do you have indexed shifters? I know the pics you had showed barends, but for some reason I thought barends were indexed. Those would help
And yes, you can do big changes. I generally use about three different gears when I'm riding, and double or triple-tap my shifters to get in the right ring for me. I'm weird in that I basically am always in the middle ring on my Trek FX, in fact I've never actually used the small ring on it! But I find that it does very well at avoiding cross chaining, and by sticking to the middle ring I can avoid trying to kill myself by mashing on the 48x11 big gear
. Deraileurs seem to be pretty resistant to abuse, I'm notorious for getting all worked up and slamming mine into gears... generally with a very solid "CLANG" from a ring
. This is hard on components, though, and should be avoided.
Basically, just find a good rhythm and system for you. I know people that shift constantly, and some who will plant themselves in a gear when they have gotten moving. Of course everyone criticizes every technique, but if it works for you I say go for it!