Just Started Out

[Racing roadkill]

That's just experience, you'll learn to plan your gear changes better as you ride more. When changing down always go up one or two at the back too.

I'm 6'5 and not that much lighter than you, I found a huge difference moving to a 32t rear cassette, with a compact crank.
But ultimately the only way to get better on hills is to do hills. I've started going out of my way to find the steepest drags on my commute home and i've really noticed the difference.

This is very true. The more hills you do, the better you get at hills. There's no real substitute. I do know a few folk who have switched to big sprockets ( 30t + ) a bit too early. The big sprockets are very welcome / necessary if you're thinking of doing a tour somewhere mountainous, but a bit overkill, for a sub 1 mile kick up a sub 10 percent gradient I.M.O. However, it is each to their own, and if it keeps people riding, then so be it.

 

[Nigelnaturist]

I do know a few folk who have switched to big sprockets ( 30t + ) a bit too early. T.

Depends what you have on the front. Not everyone one rides a ring as small as a 34 front and changing the rear is easier than the front (usually) example
38x30=33.6"
34x30=30"
My setup if its a 50/38/28 and 12/23 my lowest would be 32.3" and trust me i know at least one hill I have to work hard on in that gear. Though my current setup of 50/38/26 12-27 goes as low as 25.5" (figures based on 700x25c)