I like playing around with gear shifts on my hybrid, but I'm probably doing something wrong. When shifting to an easier gear, the sudden loss of weight/resistance and momentum surprises me and I usually freeze/coast for a bit until the resistance returns to the pedals. This is not a problem on flats, but it usually bites me back on hills and inclines. The pedal resistance and cadence is just right at the first quarter of the incline, but become very heavy by mid-hill.

What does a proper gear shift-down feel like? How does it feel being in the proper gear on a hill climb?

Downshift/shift to the "easier gear" just before you hit an incline.

Everybody develops their own style of climbing a hill. When approaching a long or steep hill, I shift down to my small chainring, but shift up a gear or two in back and try to keep my cadence up. My goal is to keep a steady cadence up the climb by using the remaining back gears (usually 4 or 5). The key is to get into a rhythm and stick with it even if it starts to hurt a little.

You didn't state the size of your chain rings & cassette.
I ASSUME, you have one of the 11-3?T cassettes?
They tend to have rather wide gaps.
IF you can use a smaller largest cog, a different cassette will give smaller "jumps" between gears.
IOW, IF you NEVER use the smallest chain ring, swap the cassette and then use the smallest CR on hills.

Just figure out the smallest gear ratio that you'll need, and then calculate that ratio with your smallest ring and largest needed cog. IF you end up with a number that isn't available, use the next larger size cog.

I assume you have a 7-speed or more and that the differences between gears is not dramatic. If so, then I think your issue is in cadence. Find a cadence that is quick and comfortable to you and shift frequently to maintain that cadence. When approaching a hill, shift a little before you need to. Finally, be cognizant of which gear you're in so that you're using your front and rear gears in close harmony. I have a 9-speed with a triple front and my front-rear shifting pattern is this: 3-9; 3-8; 3-7; 2-7; 2-6; 2-5; 2-4; 2-3; 1-3; 1-2; 1-1. Shifting into gears outside this pattern is extraneous and sometimes counter-productive.

This. But you need to practice (with different grades) to be able to determine the proper ratio.

I have a 9-speed with triple front as well. Thanks for your shifting pattern sample. I guess my problem is a combination of poor pedal stroke and poor choice of gears which is why I can't maintain a good cadence (it drops to half once I reach an incline). I'd like to put the blame on my cranks which are 15 mm too long for my height, but the engine probably has a lot to do with it as well!

It sounds to me like you should shift a little earlier if your cadence is too low. Sometimes earlier is better. If you wait until you have to shift, you are usually at a point that soft pedaling for a smoother shift isn't as fluid. Some people shift too early, some too late, and as far as I know there is no magic number to help us figure out the best point for each of us. It takes practice to find the specific shift points and methods that will work for you. For me, even a different grade makes things a bit different.

You may also want to spend some time spinning in a gear that feels too easy periodically during the flat portions of your ride in order to gradually increase the top cadence you are comfortable with. This could help you feel a little more comfortable with the changes after a down-shift.

Cranks too long can impact your ability to spin well, because your feet have to travel further with each revolution... But, some prefer longer or shorter cranks for various reasons. Short is good for higher cadence, and they help for some of us with knee issues, and of course shorter riders. Mountain bike cranks have always been relatively long for the extra leverage at low rpm... Some of this mindset seems to make it into the Hybrid world on certain bikes as well.

As far as shifting patterns, they work really great for some people, I especially like the one mentioned above (for my taste), but I also know people that print gearing charts to tape to their top tubes and they always know just how to shift to keep their cadence optimized by shifting a combination of front and back shifts. Sheldon Brown's gear calculator creates grids that would be suitable for that.

However, there are alternatives. I personally find I do better by simplifying a bit. I think it works for me partly because I have conditioned myself to be relatively comfortable at a wider cadence range. In my method, I rarely shift the front. I see each front shift as getting me in the right "range" for the riding I plan to do, and don't shift until the terrain changes significantly.

Your choices may differ, but the following works for me (in general), and I adjust depending on terrain, surface etc

Front high: For down hills, or flats with a tail wind.
Front Middle gear: For level ground and slight climbs
Front Low gear: For significant climbs

I shift the front down as I approach a climb where I will need a lower range, and as I shift down on the front, I shift the back up a couple of gears to keep my cadence and momentum... Then shift down as I need to during the climb.

On a 30 mile ride, I sometimes don't shift the front at all (rail trails and other very flat rides), and use it very seldom on hillier road rides.

Good luck in finding the approach you are most comfortable with.

timing is everything