Can I change my rear cassette to get lower gearing?

[Crankin]

So, one of the things I usually teach people, that is as you approach a hill, especially one you don't "know," is to gear *up* 1-2 gears, or even 3, on the rear. You will slow, but then, you are in a position to gear down on the front, asap. You are really cross chaining if you are in the lowest cog and in the middle ring. When you feel you need to get down to the lowest cog in the middle ring, that's your cue to stay in a bigger cog and you get into the granny then.
Another thing is, if you are dropping your chain, check your limiter screw on the derailleur (your DH can probably do this). But dropping the chain is usually a signal you are in a cross chained combo.

[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.