If you are intent on having that bike & don't care about keeping it original, hill climbing would be facilitated by lightening it up. But that means, removing some of the parts that are on there & getting it down to the basics. You are packing around a lot of extra weight in the pics showing. It shouldn't cost you all that much to do either...Just grab a wrench & have at it! Removing fenders, the lighting, kick-stand, kryptonite lock, racks, seatpost, 2 chain guards, & extra brake levers (green) will lighten it up considerably. Replacing the freewheel to a lower gear (bigger cog size) & toe clips or step-in pedals will also aid when climbing. All these improvements should ease climbing.

Better brake levers & cables would be an upgrade for what you're using now. The Weinmann center-pull brakes on there are sufficient enough to handle braking (new pads?). Good tires & wheels are vital for high speed thrills. 35 mph is not all that fast & that bike should be able to handle it, if set-up properly. I think responsiveness/handling/stability will prove beneficial...making these changes. Not sure what size tires are on your bike, but if you don't feel comfortable w/ skinnier tires, then stay w/ what feels right & safe enough 4U. Of course, all this would be pointless if you really need all the accessories that are presently on the bike.

OTOH: Getting a better bike to suit you will give you whole new perspective on your riding. :>)