Can I change my rear cassette to get lower gearing?

[ny biker]

I have never learned the "right way" to shift. With the triple, I would go up hills in the middle chain ring and whatever cog worked. If I shifted down to the easiest cog and needed an easier gear, I shifted to the small chain ring without changing anything in back. I don't shift up when I stand on the pedals, either.

[Crankin]

I never stand, except to stretch. It reduces my speed so much and hurts so much, I have developed into a "spinner." I may be going 3 mph, but I can stay seated on almost anything. I say almost, because there have been two 17-20% grade climbs in the Berkshires that had me off my bike. It still kills me I had to walk, as I have some pride. But, this is just a style preference. I know exactly when my DH will stand on a hard climb (he mostly spins, too) but it's like I cannot get myself up.
I find with beginning riders, shifting is often not intuitive. I am a spazz, but this may be the only thing about riding that I learned correctly. Sometimes on our New Members' Ride or bike rally, I coach people through on little hills.