I know this sounds painfully obvious, but if you want to become a better climber, just keep climbing. Push yourself if your knees feel happy and then climb some more.

I know people will probably argue with this, but I actually became a better climber when I got rid of my granny gear (though something that may not be an option if you have knee trouble). I felt like I was spinning like crazy, but not really going anywhere. Spinning reduces your efforts and conserves energy, so it is a great way to go uphill. But when I learned to get into the rhythm of sitting and spinning for a bit and then standing as the road really pitched up, something difficult to do in a granny gear, climbing got a whole lot easier and I was working more of my muscles. A 12x27 is now perfect for the very hilly area I live in.