I am looking to change my fork. Would changing the fork change the geometry of the bike and affect the way it handles?

perhaps, different forks have different amounts of "rake" (curve and length of the blades) and maybe the steerer tube angle. either of these will put your front wheel closer or farther away from the headtube which will change the handling.

out of curiosity, why are changing the fork? is it broken? going to a suspension fork?

it depends on fork travel, if you change the fork with different travel, this will be change your bike geometry

Meh, not particularly. Look at axle-crown measurements for suspension forks, not travel. Rigid forks have all sorts of different shapes, And I don't use them, so no idear.

True story.

Wait....so, that's the only measurement I should concern myself with? Is it that simple? No rake measurements or anything like that?

If it's a suspension fork, Yea all the rake is in the headtube angle. On rigid forks, they are sometimes raked (my '92 Schwinn Hurricane is), but someone older than me is going to have to answer specific questions about that. Personally, I like to replace forks with something as close to identical as stock as I can get.

The more rake the faster the steering, the longer the axle to crown measurement the slower the steering.

Let's stick with just suspension forks for now. Someone school me on bicycle geometry....specifically, head tube angle and the way it affects the ride.

My current (80mm) fork has a 430mm axle to crown measurement.
If my new fork has a 480mm axle to crown measurement, how does that affect the handling?

I would assume, that would change the headtube angle and make it less aggressive (say from 72 degrees to 70 degrees), when that happens, I'm assuming the bike won't be as "snappy"? The steering would be a little more sluggish? And riding hands free would be a lot easier? And ride height would increase (in the front?)

Would my assumptions be correct?