One thing that might help if you want to take an analytical approach to it is to map out your gear development and find your overlaps. You can get these by going to one of the many online gear calculators and inputting the number of teeth in your gears.
Try to hit those overlaps during shifting so as to minimise cross-gearing. Use the overlaps as the transition points for the front chain rings. You'll want to hit the gear sequences that allow you to transition out of the big ring before you start the incline. Once in the inner/middle or small, you'll find it easier to continue downshifting. If you've done it right, you will have transitioned out of the big ring before you're heavily crossed. Once crossed, you run the risk of dropping if you attempt a front shift with the chain heavily loaded at a high deflection angle. And as a previous poster has mentioned, some drivetrains don't take well to double-shifting so you might have to give yourself more time for the shift sequences.
Practice riding around on level ground and shifting within the overlaps to improve your practice with sequencing through those gears. When done right, you should notice very little change in effort and RPM as you go from one combo to another. Simulate the approach to a hill by transitioning out of the big ring while at the same time maintaining your ratio by upshifting the rear.